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

TL;DR

This paper analyzes the convergence properties of a two-block Gibbs sampler for Bayesian probit linear mixed models with improper priors, establishing geometric ergodicity and proposing an improved Haar PX-DA algorithm.

Contribution

It proves geometric ergodicity of the Gibbs sampler for these models and introduces a Haar PX-DA algorithm with similar computational cost, enhancing theoretical understanding.

Findings

Proved geometric ergodicity under certain conditions.

Established conditions for posterior propriety.

Developed a Haar PX-DA algorithm with comparable efficiency.

Abstract

In this article, we consider Markov chain Monte Carlo(MCMC) algorithms for exploring the intractable posterior density associated with Bayesian probit linear mixed models under improper priors on the regression coefficients and variance components. In particular, we construct the two-block Gibbs sampler using the data augmentation (DA) techniques. Furthermore, we prove geometric ergodicity of the Gibbs sampler, which is the foundation for building central limit theorems for MCMC based estimators and subsequent inferences. The conditions for geometric convergence are similar to those guaranteeing posterior propriety. We also provide conditions for posterior propriety when the design matrices take commonly observed forms. In general, the Haar parameter expansion for DA (PX- DA) algorithm is an improvement of the DA algorithm and it has been shown that it is theoretically at least as good as the DA algorithm. Here we construct a Haar PX-DA algorithm, which has essentially the same computational cost as the two-block Gibbs sampler.