On the correspondence from Bayesian log-linear modelling to logistic regression modelling with $g$-priors

TL;DR

This paper establishes a theoretical and practical link between Bayesian log-linear models and logistic regression models with g-priors, enabling inference translation between the two frameworks.

Contribution

It proves that g-priors on log-linear models induce corresponding g-priors on logistic regression parameters, extending to posterior distributions.

Findings

The correspondence holds asymptotically and in finite samples with numerical support.

Inference about main effects and interactions can be transferred between models.

The results facilitate Bayesian analysis in categorical data modeling.

Abstract

Consider a set of categorical variables where at least one of them is binary. The log-linear model that describes the counts in the resulting contingency table implies a specific logistic regression model, with the binary variable as the outcome. Within the Bayesian framework, the $g$-prior and mixtures of $g$-priors are commonly assigned to the parameters of a generalized linear model. We prove that assigning a $g$-prior (or a mixture of $g$-priors) to the parameters of a certain log-linear model designates a $g$-prior (or a mixture of $g$-priors) on the parameters of the corresponding logistic regression. By deriving an asymptotic result, and with numerical illustrations, we demonstrate that when a $g$-prior is adopted, this correspondence extends to the posterior distribution of the model parameters. Thus, it is valid to translate inferences from fitting a log-linear model to inferences within the logistic regression framework, with regard to the presence of main effects and interaction terms.