### Approximate Posterior Inference for Multiple Testing using a Hierarchical Mixed-effect Poisson Regression Model

#### Abstract

We present an approximate posterior inference methodology for a Bayesian hierarchical mixed-effect Poisson regression model. The model serves us to address the multiple testing problem in the presence of many group or cluster effects. This is carried out through a specialized Bayesian false discovery rate procedure.

The likelihood is simplified by an approximation based on Laplace's approximation for integrals and a trace approximation for the determinants. The posterior marginals are estimated using this approximated likelihood. In particular, we obtain credible regions for the parameters, as well as probability estimates for the difference between risks (Poisson intensities) associated with different groups or clusters, or different levels of the fixed effects. The methodology is illustrated through an application to a vaccine trial.

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