Reconsidering Dependency Networks from an Information Geometry Perspective

TL;DR

This paper offers an information-geometric perspective on dependency networks, analyzing their stationary distributions and proposing new learning algorithms with proven convergence properties.

Contribution

It introduces an information-geometric framework for dependency networks, including the full conditional divergence and a convergence proof for the learning process.

Findings

The upper bound on the stationary distribution is tight in practice.

The proposed learning algorithms decompose into independent subproblems.

Model distribution converges to the true distribution as data increases.

Abstract

Dependency networks (Heckerman et al., 2000) provide a flexible framework for modeling complex systems with many variables by combining independently learned local conditional distributions through pseudo-Gibbs sampling. Despite their computational advantages over Bayesian and Markov networks, the theoretical foundations of dependency networks remain incomplete, primarily because their model distributions -- defined as stationary distributions of pseudo-Gibbs sampling -- lack closed-form expressions. This paper develops an information-geometric analysis of pseudo-Gibbs sampling, interpreting each sampling step as an m-projection onto a full conditional manifold. Building on this interpretation, we introduce the full conditional divergence and derive an upper bound that characterizes the location of the stationary distribution in the space of probability distributions. We then reformulate both structure and parameter learning as optimization problems that decompose into independent subproblems for each node, and prove that the learned model distribution converges to the true underlying distribution as the number of training samples grows to infinity. Experiments confirm that the proposed upper bound is tight in practice.