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Detection of Unobserved Common Causes based on NML Code in Discrete, Mixed, and Continuous Variables (2403.06499v1)

Published 11 Mar 2024 in stat.ML, cs.IT, cs.LG, and math.IT

Abstract: Causal discovery in the presence of unobserved common causes from observational data only is a crucial but challenging problem. We categorize all possible causal relationships between two random variables into the following four categories and aim to identify one from observed data: two cases in which either of the direct causality exists, a case that variables are independent, and a case that variables are confounded by latent confounders. Although existing methods have been proposed to tackle this problem, they require unobserved variables to satisfy assumptions on the form of their equation models. In our previous study (Kobayashi et al., 2022), the first causal discovery method without such assumptions is proposed for discrete data and named CLOUD. Using Normalized Maximum Likelihood (NML) Code, CLOUD selects a model that yields the minimum codelength of the observed data from a set of model candidates. This paper extends CLOUD to apply for various data types across discrete, mixed, and continuous. We not only performed theoretical analysis to show the consistency of CLOUD in terms of the model selection, but also demonstrated that CLOUD is more effective than existing methods in inferring causal relationships by extensive experiments on both synthetic and real-world data.

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References (38)
  1. Budhathoki K, Vreeken J (2017) MDL for causal inference on discrete data. In: 2017 IEEE International Conference on Data Mining (ICDM), pp 751–756 Budhathoki and Vreeken [2018] Budhathoki K, Vreeken J (2018) Accurate causal inference on discrete data. In: 2018 IEEE International Conference on Data Mining (ICDM), pp 881–886 Choi and Ni [2023] Choi J, Ni Y (2023) Model-based causal discovery for zero-inflated count data. Journal of Machine Learning Research 24(200):1–32 Hirai and Yamanishi [2013] Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Budhathoki K, Vreeken J (2018) Accurate causal inference on discrete data. In: 2018 IEEE International Conference on Data Mining (ICDM), pp 881–886 Choi and Ni [2023] Choi J, Ni Y (2023) Model-based causal discovery for zero-inflated count data. Journal of Machine Learning Research 24(200):1–32 Hirai and Yamanishi [2013] Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Choi J, Ni Y (2023) Model-based causal discovery for zero-inflated count data. Journal of Machine Learning Research 24(200):1–32 Hirai and Yamanishi [2013] Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. 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Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Choi J, Ni Y (2023) Model-based causal discovery for zero-inflated count data. Journal of Machine Learning Research 24(200):1–32 Hirai and Yamanishi [2013] Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  3. Choi J, Ni Y (2023) Model-based causal discovery for zero-inflated count data. Journal of Machine Learning Research 24(200):1–32 Hirai and Yamanishi [2013] Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hirai S, Yamanishi K (2013) Efficient computation of normalized maximum likelihood codes for gaussian mixture models with its applications to clustering. IEEE Transactions on Information Theory 59(11):7718–7727 Hoyer et al [2008a] Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. 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The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer P, Janzing D, Mooij JM, et al (2008a) Nonlinear causal discovery with additive noise models. Advances in neural information processing systems 21 Hoyer et al [2008b] Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. 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The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Hoyer PO, Shimizu S, Kerminen AJ, et al (2008b) Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning 49(2):362–378 Janzing and Schölkopf [2010] Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. 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Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. 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In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. 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The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  7. Janzing D, Schölkopf B (2010) Causal inference using the algorithmic Markov condition. IEEE Transactions on Information Theory 56(10):5168–5194 Kaltenpoth and Vreeken [2019] Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kaltenpoth D, Vreeken J (2019) We are not your real parents: Telling causal from confounded using mdl. In: Proceedings of the 2019 SIAM International Conference on Data Mining, SIAM, pp 199–207, URL https://github.com/davidkwca/CoCa Kobayashi et al [2022] Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. 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The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kobayashi M, Miyaguchi K, Matsushima S (2022) Detection of unobserved common cause in discrete data based on the mdl principle. In: 2022 IEEE International Conference on Big Data (Big Data), IEEE, pp 45–54 Kocaoglu et al [2017] Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
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In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kocaoglu M, Dimakis AG, Vishwanath S, et al (2017) Entropic causal inference. In: Thirty-First AAAI Conference on Artificial Intelligence Kontkanen and Myllymäki [2007] Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. 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The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. 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In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. 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Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Myllymäki P (2007) A linear-time algorithm for computing the multinomial stochastic complexity. Information Processing Letters 103(6):227–233 Kontkanen et al [2008] Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
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In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Kontkanen P, Wettig H, Myllymäki P (2008) NML computation algorithms for tree-structured multinomial Bayesian networks. EURASIP Journal on Bioinformatics and Systems Biology 2007:1–11 Li et al [2022] Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. 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In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Li Y, Xia R, Liu C, et al (2022) A hybrid causal structure learning algorithm for mixed-type data. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp 7435–7443, URL https://github.com/DAMO-DI-ML/AAAI2022-HCM Liu and Chan [2016] Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. 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In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Liu F, Chan L (2016) Causal inference on discrete data via estimating distance correlations. Neural computation 28(5):801–814 Maeda and Shimizu [2020] Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. 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The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. 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In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. 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Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  15. Maeda TN, Shimizu S (2020) Rcd: Repetitive causal discovery of linear non-gaussian acyclic models with latent confounders. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp 735–745 Maeda and Shimizu [2021] Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Maeda TN, Shimizu S (2021) Causal additive models with unobserved variables. In: Uncertainty in Artificial Intelligence, PMLR, pp 97–106 Marx and Vreeken [2021] Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. 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Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. 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The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. 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The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Marx A, Vreeken J (2021) Formally justifying mdl-based inference of cause and effect. arXiv preprint arXiv:210501902 Mooij et al [2016] Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. 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Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  18. Mooij JM, Peters J, Janzing D, et al (2016) Distinguishing cause from effect using observational data: methods and benchmarks. The Journal of Machine Learning Research 17(1):1103–1204 Pearl [2009] Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  19. Pearl J (2009) Causality. Cambridge university press Perrin et al [2003] Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Perrin BE, Ralaivola L, Mazurie A, et al (2003) Gene networks inference using dynamic bayesian networks. Bioinformatics-Oxford 19(2):138–148 Peters et al [2010] Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. 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The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. 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Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. 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In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. 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  21. Peters J, Janzing D, Schölkopf B (2010) Identifying cause and effect on discrete data using additive noise models. In: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp 597–604 Peters et al [2011] Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  22. Peters J, Janzing D, Scholkopf B (2011) Causal inference on discrete data using additive noise models. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12):2436–2450 Peters et al [2014] Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  23. Peters J, Mooij JM, Janzing D, et al (2014) Causal discovery with continuous additive noise models. Journal of Machine Learning Research 15(58):2009–2053 Peters et al [2017] Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  24. Peters J, Janzing D, Schölkopf B (2017) Elements of causal inference: foundations and learning algorithms. The MIT Press Rissanen [1978] Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  25. Rissanen J (1978) Modeling by shortest data description. Automatica 14(5):465–471 Rissanen [1983] Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  26. Rissanen J (1983) A universal prior for integers and estimation by minimum description length. The Annals of statistics 11(2):416–431 Rissanen [1989] Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  27. Rissanen J (1989) Stochastic complexity in statistical inquiry. World Scientific Rissanen [2012] Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Rissanen J (2012) Optimal Parameter Estimation. Cambridge University Press Ronen et al [2002] Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
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In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  29. Ronen M, Rosenberg R, Shraiman BI, et al (2002) Assigning numbers to the arrows: parameterizing a gene regulation network by using accurate expression kinetics. Proceedings of the national academy of sciences 99(16):10555–10560 Schölkopf [2022] Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Schölkopf B (2022) Causality for machine learning. In: Probabilistic and Causal Inference: The Works of Judea Pearl. p 765–804 Shimizu et al [2006] Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. 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Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. 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  31. Shimizu S, Hoyer PO, Hyvärinen A, et al (2006) A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research 7(10) Shimizu et al [2011] Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Shimizu S, Inazumi T, Sogawa Y, et al (2011) Directlingam: A direct method for learning a linear non-gaussian structural equation model. The Journal of Machine Learning Research 12:1225–1248 Spirtes et al [2000] Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. 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In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. 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  33. Spirtes P, Glymour CN, Scheines R (2000) Causation, prediction, and search. MIT press Stegle et al [2010] Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  34. Stegle O, Janzing D, Zhang K, et al (2010) Probabilistic latent variable models for distinguishing between cause and effect. Advances in neural information processing systems 23 Tagasovska et al [2020] Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  35. Tagasovska N, Chavez-Demoulin V, Vatter T (2020) Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery. In: International Conference on Machine Learning, PMLR, pp 9311–9323 Tashiro et al [2014] Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  36. Tashiro T, Shimizu S, Hyvärinen A, et al (2014) Parcelingam: A causal ordering method robust against latent confounders. Neural computation 26(1):57–83 Xu et al [2022] Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  37. Xu S, Mian OA, Marx A, et al (2022) Inferring cause and effect in the presence of heteroscedastic noise. In: International Conference on Machine Learning, PMLR, pp 24615–24630 Zeng et al [2022] Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009 Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
  38. Zeng Y, Shimizu S, Matsui H, et al (2022) Causal discovery for linear mixed data. In: Conference on Causal Learning and Reasoning, PMLR, pp 994–1009
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Authors (3)
  1. Masatoshi Kobayashi (2 papers)
  2. Kohei Miyagichi (1 paper)
  3. Shin Matsushima (8 papers)

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