Causal inference using the algorithmic Markov condition (0804.3678v1)

Abstract: Inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure Markovian. Here we argue why causal inference is also possible when only single observations are present. We develop a theory how to generate causal graphs explaining similarities between single objects. To this end, we replace the notion of conditional stochastic independence in the causal Markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules. We explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities, making it possible to select among Markov equivalent causal graphs. This insight provides a theoretical foundation of a heuristic principle proposed in earlier work. We also discuss how to replace Kolmogorov complexity with decidable complexity criteria. This can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution.

• The paper presents a novel causal inference framework leveraging algorithmic mutual information to identify causal links from single observations.
• The paper develops practical complexity criteria to navigate the undecidability of Kolmogorov complexity and distinguish true causal dependencies.
• The paper establishes new statistical inference rules that minimize algorithmic dependence among Markov kernels to validate model independence.

Algorithmic Markov Condition for Causal Inference

Dominik Janzing and Bernhard Schölkopf's paper presents a sophisticated approach to causal inference, leveraging the principles of the algorithmic Markov condition to draw causal connections from both statistical data and individual observations. The paper delineates a theoretical framework that embarks on a departure from traditional statistical models, utilizing insights from algorithmic information theory to enhance our understanding of causality.

Key Contributions

1. Algorithmic Markov Condition: The authors introduce a novel causal inference framework based on the algorithmic Markov condition, which extends the classical causal Markov condition by incorporating algorithmic mutual information. This reformulation offers a tool for assessing causal relations by considering the minimal algorithmic dependencies among individual observations.
2. Inference from Single Observations: A pivotal claim is the possibility of causal inference from single observations rather than relying on repeated i.i.d. sampling. This perspective is grounded in comparing the Kolmogorov complexities of individual objects to detect causal links, thus broadening the applicability of causal reasoning to scenarios lacking abundant data or scenarios dealing with high-complexity objects.
3. Decidable Complexity Criteria: The paper acknowledges the inherent undecidability associated with Kolmogorov complexity and proposes practical, decidable modifications for inference, thus bridging the gap between theoretical insights and empirical applications. These include exploiting symmetry constraints and resource-bounded complexity to approximate the entitled complexities.
4. Novel Statistical Inference Rules: The theoretical groundwork laid by the algorithmic Markov condition yields new statistical inference rules. These rules allow for discerning causal structures that minimize the total algorithmic dependence among Markov kernels, offering an additional dimension for differentiating between causal and acausal models.
5. Importance of Independence of Mechanisms: A significant assertion in the discourse is the necessity to prefer models where the statistical properties (Markov kernels) remain algorithmically independent. This stance aspires to distinguish between true causal links and those merely emerging from dependencies among statistical mechanisms.

Implications and Future Prospects

The implications of this work are substantial for both theoretical and practical domains. Theoretically, it provides a more comprehensive framework for reasoning about causality by intertwining concepts from algorithmic information theory with causal inference. Practically, the proposed methods could improve causal discovery in settings with limited data, where traditional statistical inference falls short.

Moreover, the paper hints at the potential for Bayesian approximations to causal inference, where priors could be constructed over possible causal models to guide the learning process. This opens avenues for future work in integrating Bayesian methods with the principles outlined in the algorithmic Markov condition.

Speculation on Future Developments

The integration of algorithmic information theory into causal inference marks a pronounced evolution in understanding causality. As computational capabilities grow, the resource-bounded complexities and approximations posited by Janzing and Schölkopf may become more feasible, driving advancements in both machine learning and artificial intelligence. Furthermore, investigating the interplay between algorithmic and computational complexities in causal models could reveal even more nuanced insights into causality.

In conclusion, this paper invites a reevaluation of conventional causal inference paradigms and sets the foundation for a richer, more versatile narrative on causality. The algorithmic Markov condition not only contributes to the ongoing dialogue in causal discovery but also spearheads a methodological shift towards harnessing the complexity inherent in real-world data.