Tutorial on Exact Belief Propagation in Bayesian Networks: from Messages to Algorithms

TL;DR

This paper introduces a formalism for explicitly defining messages in Bayesian networks, generalizing the approach used in hidden Markov models, and derives recursive algorithms with illustrative examples.

Contribution

It provides a new explicit message definition framework for Bayesian networks, enabling clearer derivation of algorithms and properties, with practical examples and code.

Findings

Explicit message definitions facilitate derivation of belief propagation algorithms.

The formalism applies to complex Bayesian network structures like pedigrees.

Standalone R code supports practical implementation.

Abstract

In Bayesian networks, exact belief propagation is achieved through message passing algorithms. These algorithms (ex: inward and outward) provide only a recursive definition of the corresponding messages. In contrast, when working on hidden Markov models and variants, one classically first defines explicitly these messages (forward and backward quantities), and then derive all results and algorithms. In this paper, we generalize the hidden Markov model approach by introducing an explicit definition of the messages in Bayesian networks, from which we derive all the relevant properties and results including the recursive algorithms that allow to compute these messages. Two didactic examples (the precipitation hidden Markov model and the pedigree Bayesian network) are considered along the paper to illustrate the new formalism and standalone R source code is provided in the appendix.