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