A Bayesian Approach to Learning Bayesian Networks with Local Structure (1302.1528v2)

Abstract: Recently several researchers have investigated techniques for using data to learn Bayesian networks containing compact representations for the conditional probability distributions (CPDs) stored at each node. The majority of this work has concentrated on using decision-tree representations for the CPDs. In addition, researchers typically apply non-Bayesian (or asymptotically Bayesian) scoring functions such as MDL to evaluate the goodness-of-fit of networks to the data. In this paper we investigate a Bayesian approach to learning Bayesian networks that contain the more general decision-graph representations of the CPDs. First, we describe how to evaluate the posterior probability that is, the Bayesian score of such a network, given a database of observed cases. Second, we describe various search spaces that can be used, in conjunction with a scoring function and a search procedure, to identify one or more high-scoring networks. Finally, we present an experimental evaluation of the search spaces, using a greedy algorithm and a Bayesian scoring function.

• The paper introduces a Bayesian score evaluation method that computes posterior probabilities for network structures using decision graphs.
• It details efficient operators for modifying decision graphs to navigate complex search spaces in Bayesian network discovery.
• Empirical results show that decision graphs yield better model fit and more parsimonious representations of conditional probabilities.

A Bayesian Approach to Learning Bayesian Networks with Local Structure

In the paper "A Bayesian Approach to Learning Bayesian Networks with Local Structure," Chickering, Heckerman, and Meek present a refined methodology for learning Bayesian networks that deploy decision graphs to represent conditional probability distributions (CPDs). This work builds upon prior efforts which predominantly utilized decision-tree representations and often relied on non-Bayesian or asymptotically Bayesian scoring metrics, such as the Minimum Description Length (MDL), to measure model fit to data.

Core Contributions

The authors propose a novel Bayesian method for learning Bayesian networks that incorporate decision graphs, an extension of decision trees capable of encoding complex equality constraints among parameters. The core contributions of the paper include:

1. Bayesian Score Evaluation: The authors detail a process for computing the posterior probability or the Bayesian score for a network structure that integrates decision graphs. This involves comprehensive assumptions such as parameter independence and Dirichlet priors, facilitating closed-form derivation of scores.
2. Search Spaces for Network Discovery: Various search spaces are explored that integrate scoring functions and search procedures to identify high-scoring networks. They define operators for modifying decision graphs within a node, such as complete splits, binary splits, and merges, intending to navigate complex search spaces efficiently.
3. Empirical Evaluation: The paper evaluates their method through experiments demonstrating that decision graphs often lead to better-fitted Bayesian network models compared to structures constrained to decision trees or complete tables.

Noteworthy Insights and Results

• The empirical investigations indicate a noticeable improvement, particularly in terms of the posterior probabilities of the identified structures when using decision graphs compared to using decision trees. This suggests significant gains in leveraging decision graphs for capturing CPD structures.
• An interesting aspect discussed is the sufficiency of operators (such as complete split and merge) to move between different sets of parameter constraints, highlighting the expressive strength of decision graphs.
• The authors show that decision graphs can express parameter constraints that would yield independence conditions, facilitating more parsimonious network structures and potentially reducing overfitting.

Implications and Future Directions

This research has several implications:

• Practical Utilization: This methodology could be substantially useful in domains where computational tractability is a concern, and decision graphs allow for a reduction in parameterization complexity while still maintaining the flexibility for modeling intricate dependencies.
• Theoretical Advancement: There is room for future research to explore decision graphs that can incorporate local nodes splitting on themselves or other novel arrangements leading to potentially more expressive models. This points to possible exploration of non-Dirichlet priors or relaxations on parameter independence assumptions to handle a broader class of graphical models.
• Algorithmic Complexity: The work delineates a greedy search paradigm, which, while demonstrating improvements, opens questions regarding optimizations or alternative search methods such as stochastic or evolutionary algorithms that might explore the search space more effectively or avoid local optima inherent in greedy approaches.

Chickering, Heckerman, and Meek's research provides a structured yet flexible framework for learning Bayesian networks with enriched local structures, paving avenues for practical applications and theoretical explorations within artificial intelligence and machine learning. Their approach reaffirms the importance of Bayesian methods in learning probabilistic models but also invites further refinements and adaptations to meet diverse, real-world data challenges.