Gibbs Sampling for a Bayesian Hierarchical Version of the General Linear Mixed Model

We consider two-component block Gibbs sampling for a Bayesian hierarchical version of the normal theory general linear mixed model. This model is practically relevant in the sense that it is general enough to have many applications and in that it is not straightforward to sample directly from the corresponding posterior distribution. There are two possible orders in which to update the components of our block Gibbs sampler. For both update orders, drift and minorization conditions are constructed for the corresponding Markov chains. Most importantly, these results establish geometric ergodicity for the block Gibbs sampler. We also construct a minorization condition that will allow practitioners to exploit regenerative simulation techniques for constructing a reasonable initial distribution and constructing valid Monte Carlo standard errors.