[Paper Review] The Bayesian Structural EM Algorithm

Nir Friedman|arXiv (Cornell University)|Jan 30, 2013

Bayesian Modeling and Causal InferenceComputer Science22 references516 citations

TL;DR

This paper introduces the Bayesian Structural EM algorithm, a novel method that integrates Bayesian model selection with the Structural EM framework to learn Bayesian network structures from incomplete data. By combining expectation-maximization for parameter estimation with score-based structure search using Bayesian scores, the algorithm ensures convergence and enables effective learning of probabilistic models even with missing or hidden variables.

ABSTRACT

In recent years there has been a flurry of works on learning Bayesian networks from data. One of the hard problems in this area is how to effectively learn the structure of a belief network from incomplete data- that is, in the presence of missing values or hidden variables. In a recent paper, I introduced an algorithm called Structural EM that combines the standard Expectation Maximization (EM) algorithm, which optimizes parameters, with structure search for model selection. That algorithm learns networks based on penalized likelihood scores, which include the BIC/MDL score and various approximations to the Bayesian score. In this paper, I extend Structural EM to deal directly with Bayesian model selection. I prove the convergence of the resulting algorithm and show how to apply it for learning a large class of probabilistic models, including Bayesian networks and some variants thereof.

Motivation & Objective

• Address the challenge of learning Bayesian network structures from data with missing values or hidden variables.
• Overcome limitations of traditional score-based methods that rely on penalized likelihood (e.g., BIC/MDL) and do not fully account for prior knowledge in model selection.
• Develop a unified framework that integrates parameter learning (via EM) with structure learning under full Bayesian scoring.
• Ensure theoretical convergence of the learning process in the presence of latent variables.
• Extend the applicability of the Structural EM algorithm to a broader class of probabilistic models beyond standard Bayesian networks.

Proposed method

• Adapt the Structural EM algorithm to use Bayesian scores instead of penalized likelihood scores for model selection.
• Integrate the Expectation-Maximization (EM) algorithm for parameter estimation in the presence of missing or hidden variables.
• Perform iterative optimization: alternate between E-step (computing expected sufficient statistics under current model) and M-step (updating parameters and structure).
• Use marginal likelihood (Bayesian score) as the objective function for structure search, incorporating prior distributions over network structures.
• Apply a greedy or heuristic search strategy over the space of DAGs, guided by the Bayesian score, to identify high-scoring structures.
• Prove convergence of the algorithm under mild regularity conditions, ensuring stable learning dynamics.

Experimental results

Research questions

• RQ1 Can Bayesian model selection be effectively integrated into the Structural EM framework for learning Bayesian networks with incomplete data?
• RQ2 How does the use of Bayesian scores improve structure learning compared to penalized likelihood scores like BIC/MDL in the presence of hidden variables?
• RQ3 Does the proposed algorithm guarantee convergence when learning both parameters and structure simultaneously?
• RQ4 To what extent can the algorithm handle complex probabilistic models beyond standard Bayesian networks?
• RQ5 What is the impact of prior distributions on the structure learning process under incomplete data?

Key findings

• The Bayesian Structural EM algorithm converges to a local optimum under mild regularity conditions, ensuring stable learning dynamics.
• The algorithm enables effective structure learning in the presence of missing or hidden variables by leveraging full Bayesian scoring.
• By replacing penalized likelihood scores with Bayesian scores, the method incorporates prior knowledge and improves model selection accuracy.
• The framework is general enough to be applied to a wide class of probabilistic models, including Bayesian networks and their variants.
• Theoretical analysis confirms that the iterative optimization process maintains monotonic improvement in the Bayesian score, leading to reliable convergence.