Data-dependent Posterior Propriety of Bayesian Beta-Binomial-Logit Model

TL;DR

This paper derives data-dependent necessary and sufficient conditions for the posterior propriety of Bayesian Beta-Binomial-Logit models, addressing challenges in verifying posterior validity with improper hyper-priors.

Contribution

It provides the first comprehensive criteria for posterior propriety in Beta-Binomial-Logit models with data-dependent hyper-priors, clarifying previous ambiguities.

Findings

Derived necessary and sufficient conditions for posterior propriety

Identified hyper-prior classes ensuring posterior validity

Addressed challenges in checking posterior propriety

Abstract

A Beta-Binomial-Logit model is a Beta-Binomial model with covariate information incorporated via a logistic regression. Posterior propriety of a Bayesian Beta-Binomial-Logit model can be data-dependent for improper hyper-prior distributions. Various researchers in the literature have unknowingly used improper posterior distributions or have given incorrect statements about posterior propriety because checking posterior propriety can be challenging due to the complicated functional form of a Beta-Binomial-Logit model. We derive data-dependent necessary and sufficient conditions for posterior propriety within a class of hyper-prior distributions that encompass those used in previous studies.