Convergence analysis of the Gibbs sampler for Bayesian general linear mixed models with improper priors

TL;DR

This paper analyzes the convergence rate of the Gibbs sampler used in Bayesian general linear mixed models with improper priors, providing practical conditions for geometric ergodicity and enabling valid inference.

Contribution

It offers a simple, checkable condition for the geometric ergodicity of the Gibbs sampler in this context, close to the minimal requirement for posterior propriety.

Findings

Provides a sufficient condition for geometric ergodicity

Ensures the existence of central limit theorems for estimates

Facilitates valid asymptotic standard errors

Abstract

Bayesian analysis of data from the general linear mixed model is challenging because any nontrivial prior leads to an intractable posterior density. However, if a conditionally conjugate prior density is adopted, then there is a simple Gibbs sampler that can be employed to explore the posterior density. A popular default among the conditionally conjugate priors is an improper prior that takes a product form with a flat prior on the regression parameter, and so-called power priors on each of the variance components. In this paper, a convergence rate analysis of the corresponding Gibbs sampler is undertaken. The main result is a simple, easily-checked sufficient condition for geometric ergodicity of the Gibbs-Markov chain. This result is close to the best possible result in the sense that the sufficient condition is only slightly stronger than what is required to ensure posterior propriety. The theory developed in this paper is extremely important from a practical standpoint because it guarantees the existence of central limit theorems that allow for the computation of valid asymptotic standard errors for the estimates computed using the Gibbs sampler.