A conjugate prior for discrete hierarchical log-linear models

TL;DR

This paper introduces a flexible conjugate prior for discrete hierarchical log-linear models in Bayesian analysis, generalizing existing priors and aiding in model selection for contingency tables.

Contribution

It defines a new family of conjugate priors for hierarchical log-linear models, including graphical models, with properties that facilitate Bayesian inference and model selection.

Findings

The prior generalizes the hyper Dirichlet prior.

It is conjugate under multinomial sampling.

Application to model selection in six-way tables.

Abstract

In Bayesian analysis of multi-way contingency tables, the selection of a prior distribution for either the log-linear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical log-linear models, which includes the class of graphical models. These priors are defined as the Diaconis--Ylvisaker conjugate priors on the log-linear parameters subject to "baseline constraints" under multinomial sampling. We also derive the induced prior on the cell probabilities and show that the induced prior is a generalization of the hyper Dirichlet prior. We show that this prior has several desirable properties and illustrate its usefulness by identifying the most probable decomposable, graphical and hierarchical log-linear models for a six-way contingency table.