Posterior simulation in the generalized linear mixed model with semiparametric random effects

Generalized linear mixed models with semiparametric random effects are useful in a wide variety of Bayesian applications. When the random effects arise from a mixture of Dirichlet process (MDP) model with normal base measure, Gibbs sampling algorithms based on the Pólya urn scheme are often used to simulate posterior draws in conjugate models (essentially, linear regression models and models for binary outcomes). In the non-conjugate case, the algorithms proposed by MacEachern and Müller (1998) and Neal (2000) are often applied to generate posterior samples. Some common problems associated with simulation algorithms for non-conjugate models include convergence and mixing difficulties. This paper proposes an algorithm for MDP models with exponential family likelihoods and normal base measures. The algorithm proceeds by making a Laplace approximation to the likelihood function, thereby matching the proposal with that of the Gibbs sampler. The proposal is accepted or rejected via a Metropolis-Hastings step. For conjugate MDP models, the algorithm is identical to the Gibbs sampler. The performance of the technique is investigated using a Poisson regression model with semiparametric random effects. The algorithm performs efficiently and reliably, even in problems where large sample results do not guarantee the success of the Laplace approximation. ∗Subharup Guha is Assistant Professor, Department of Statistics, University of Missouri-Columbia, Columbia, MO 65211-6100 (E-mail: [email protected]). The author thanks Professor Steven MacEachern, the unknown Associate Editor and the unknown referee for many insightful comments that helped improve the focus of the paper.