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Bayesian analysis of data from the general linear mixed model is challengingbecause any nontrivial prior leads to an intractable posterior density.However, if a conditionally conjugate prior density is adopted, then there is asimple Gibbs sampler that can be employed to explore the posterior density. Apopular default among the conditionally conjugate priors is an improper priorthat takes a product form with a flat prior on the regression parameter, andso-called power priors on each of the variance components. In this paper, aconvergence rate analysis of the corresponding Gibbs sampler is undertaken. Themain result is a simple, easily-checked sufficient condition for geometricergodicity of the Gibbs-Markov chain. This result is close to the best possibleresult in the sense that the sufficient condition is only slightly strongerthan what is required to ensure posterior propriety. The theory developed inthis paper is extremely important from a practical standpoint because itguarantees the existence of central limit theorems that allow for thecomputation of valid asymptotic standard errors for the estimates computedusing the Gibbs sampler.

Convergence analysis of the Gibbs sampler for Bayesian general linear mixed models with improper priors