Gibbs Sampling for a Bayesian Hierarchical General Linear Model

We consider two-component block Gibbs sampling for a Bayesian hierarchical version of the normal theory general linear 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 general minorization condition and use it to investigate the applicability of regenerative simulation techniques for constructing valid Monte Carlo standard errors. With these contributions, practitioners using our block Gibbs sampler can be as confident in making inference about the posterior from the resulting simulations as they would be with inferences based on random samples from the posterior.