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.