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In this article, we consider Markov chain Monte Carlo(MCMC) algorithms forexploring the intractable posterior density associated with Bayesian probitlinear mixed models under improper priors on the regression coefficients andvariance components. In particular, we construct the two-block Gibbs samplerusing the data augmentation (DA) techniques. Furthermore, we prove geometricergodicity of the Gibbs sampler, which is the foundation for building centrallimit theorems for MCMC based estimators and subsequent inferences. Theconditions for geometric convergence are similar to those guaranteeingposterior propriety. We also provide conditions for posterior propriety whenthe design matrices take commonly observed forms. In general, the Haarparameter expansion for DA (PX- DA) algorithm is an improvement of the DAalgorithm and it has been shown that it is theoretically at least as good asthe DA algorithm. Here we construct a Haar PX-DA algorithm, which hasessentially the same computational cost as the two-block Gibbs sampler.

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