Block Gibbs sampling for Bayesian random effects models with improper priors: Convergence and regeneration Aixin Tan and James P. Hobert Department of Statistics, University of Florida May 5th 2008 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 paper, theoretical and method- ological 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 simu- lation 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 paper is a result showing that, unless the data set 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. 1 Introduction Consider the classical one-way random effects model given by Y ij = θ i + ε ij ,i =1,...,q, j =1,...,m i , (1) where the random effects θ 1 ,...,θ q are iid N(μ, σ 2 θ ), the ε ij s are iid N(0,σ 2 e ) and independent of the θ i s, and (μ, σ 2 θ ,σ 2 e ) is an unknown parameter. There is a long history of Bayesian analysis using this Key words and phrases. Asymptotic variance, Central limit theorem, Convergence rate, Drift condition, Geometric ergod- icity, Minorization condition, One-way model, Standard error 1