  Bayesian analysis of data from the general linear mixed model is challenging
because any nontrivial prior leads to an intractable posterior density.
However, if a conditionally conjugate prior density is adopted, then there is a
simple Gibbs sampler that can be employed to explore the posterior density. A
popular default among the conditionally conjugate priors is an improper prior
that takes a product form with a flat prior on the regression parameter, and
so-called power priors on each of the variance components. In this paper, a
convergence rate analysis of the corresponding Gibbs sampler is undertaken. The
main result is a simple, easily-checked sufficient condition for geometric
ergodicity of the Gibbs-Markov chain. This result is close to the best possible
result in the sense that the sufficient condition is only slightly stronger
than what is required to ensure posterior propriety. The theory developed in
this paper is extremely important from a practical standpoint because it
guarantees the existence of central limit theorems that allow for the
computation of valid asymptotic standard errors for the estimates computed
using the Gibbs sampler.
