Extending Factor Graphs so as to Unify Directed and Undirected Graphical Models

TL;DR

This paper introduces factor graphs as a unified graphical model framework that combines the strengths of Bayesian networks and Markov random fields, enabling more flexible and expressive probabilistic modeling and inference.

Contribution

The authors demonstrate that factor graphs can represent all Bayesian networks and MRFs, extend to models beyond them, and facilitate a unified message-passing inference algorithm.

Findings

Factor graphs can express all Bayesian network and MRF independencies.

They can represent models that are not possible with Bayesian networks or MRFs alone.

A modified Bayes-ball algorithm is proposed for conditional independence in factor graphs.

Abstract

The two most popular types of graphical model are directed models (Bayesian networks) and undirected models (Markov random fields, or MRFs). Directed and undirected models offer complementary properties in model construction, expressing conditional independencies, expressing arbitrary factorizations of joint distributions, and formulating message-passing inference algorithms. We show that the strengths of these two representations can be combined in a single type of graphical model called a 'factor graph'. Every Bayesian network or MRF can be easily converted to a factor graph that expresses the same conditional independencies, expresses the same factorization of the joint distribution, and can be used for probabilistic inference through application of a single, simple message-passing algorithm. In contrast to chain graphs, where message-passing is implemented on a hypergraph, message-passing can be directly implemented on the factor graph. We describe a modified 'Bayes-ball' algorithm for establishing conditional independence in factor graphs, and we show that factor graphs form a strict superset of Bayesian networks and MRFs. In particular, we give an example of a commonly-used 'mixture of experts' model fragment, whose independencies cannot be represented in a Bayesian network or an MRF, but can be represented in a factor graph. We finish by giving examples of real-world problems that are not well suited to representation in Bayesian networks and MRFs, but are well-suited to representation in factor graphs.