Necessary and sufficient conditions for posterior propriety for generalized linear mixed models

TL;DR

This paper establishes clear, verifiable conditions for ensuring the posterior distribution is proper in Bayesian generalized linear mixed models, especially for binomial and Poisson cases, using improper and proper priors.

Contribution

It provides the first set of easily checkable sufficient conditions for posterior propriety in common GLMMs with various priors, including new approximate Jeffreys' priors.

Findings

Sufficient conditions for posterior propriety in binomial and Poisson GLMMs

Necessary conditions for general exponential family GLMMs

Demonstrations with one-way and two-way random effects models

Abstract

Generalized linear mixed models (GLMMs) are commonly used to analyze correlated discrete or continuous response data. In Bayesian GLMMs, the often-used improper priors may yield undesirable improper posterior distributions. Thus, verifying posterior propriety is crucial for valid applications of Bayesian GLMMs with improper priors. Here, we consider the popular improper uniform prior on the regression coefficients and several proper or improper priors, including the widely used gamma and power priors on the variance components of the random effects. We also construct an approximate Jeffreys' prior for objective Bayesian analysis of GLMMs. For the two most widely used GLMMs, namely, the binomial and Poisson GLMMs, we provide easily verifiable sufficient conditions compared to the currently available results. We also derive the necessary conditions for posterior propriety for the general exponential family GLMMs. Finally, we use examples involving one-way and two-way random effects models to demonstrate the theoretical results derived here.