On the Prior and Posterior Distributions Used in Graphical Modelling

TL;DR

This paper investigates the prior and posterior distributions over graph structures in Bayesian graphical model learning, characterizing their behavior and proposing measures of structural variability for Bayesian and Markov networks.

Contribution

It provides a detailed characterization of distributions over graph structures and introduces measures of structural variability for graphical models.

Findings

Characterization of prior and posterior distributions over graph structures

Development of measures for structural variability in graphical models

Insights into the behavior of distributions as a function of graph edges

Abstract

Graphical model learning and inference are often performed using Bayesian techniques. In particular, learning is usually performed in two separate steps. First, the graph structure is learned from the data; then the parameters of the model are estimated conditional on that graph structure. While the probability distributions involved in this second step have been studied in depth, the ones used in the first step have not been explored in as much detail. In this paper, we will study the prior and posterior distributions defined over the space of the graph structures for the purpose of learning the structure of a graphical model. In particular, we will provide a characterisation of the behaviour of those distributions as a function of the possible edges of the graph. We will then use the properties resulting from this characterisation to define measures of structural variability for both Bayesian and Markov networks, and we will point out some of their possible applications.