Block Gibbs Sampling for Bayesian Random Effects Models With Improper Priors: Convergence and Regeneration

Abstract

Bayesian versions of the classical one-way random effects model are widely used to analyze data. If the standard diffuse prior is adopted, there is a simple block Gibbs sampler that can be employed to explore the intractable posterior distribution. In this article, theoretical and methodological results are developed that allow one to use this block Gibbs sampler with the same level of confidence that one would have using classical (iid) Monte Carlo. Indeed, a regenerative simulation method is developed that yields simple, asymptotically valid standard errors for the ergodic averages that are used to estimate intractable posterior expectations. These standard errors can be used to choose an appropriate (Markov chain) Monte Carlo sample size. The regenerative method rests on the assumption that the underlying Markov chain converges to its stationary distribution at a geometric rate. Another contribution of this article is a result showing that, unless the dataset is extremely small and unbalanced, the block Gibbs Markov chain is geometrically ergodic. We illustrate the use of the regenerative method with data from a styrene exposure study. R code for the simulation is posted as an online supplement.