Geometric ergodicity of Gibbs and block Gibbs samplers for a hierarchical random effects model

We consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierar chical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Ma rkov chains are geometrically ergodic. Showing that a Gibbs sampler is geom etrically ergodic is the first step toward establishing central limit theor ems, which can be used to approximate the error associated with Monte Carlo estimates of posterior quantities of interest. Thus, our results will be o f practical interest to researchers using these Gibbs samplers for Bayesian data analysis. (C) 1998 Academic Press.