Non-Conjugate Bayesian Analysis of Variance Component Models

Russell D. Wolfinger and Robert E. Kass


We consider the usual normal linear mixed model for variance components from a Bayesian viewpoint. With conjugate priors and balanced data, Gibbs sampling is easy to implement. However, simulating from full conditionals can become difficult for the analysis of unbalanced data with possibly non-conjugate priors, thus leading one to consider alternative Markov chain Monte Carlo schemes. We propose and investigate a method for simulating from posterior distribution based on an independence chain. The method is customized to exploit the structure of the variance component model, and it works with arbitrary prior distributions. As a default reference prior, we consider a version of Jeffreys prior based on the integrated (``restricted'') likelihood. We demonstrate the ease of application and flexibility of this approach in several familiar settings, even in the presence of unbalanced data.

Keywords: Jeffreys prior, Mixed model, Posterior simulation, Reference prior, REML

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