Alternative Methods for Fitting Two-Stage Hierarchical Bayesian Models

Although it is common practice to fit a complex Bayesian model using Markov chain Monte Carlo (MCMC) methods, we provide an alternative sampling-based method to fit a two-stage hierarchical model in which there is conjugacy conditional on the parameters in the second stage. Using the sampling/importance resampling (SIR) algorithm, our method subsamples independent samples from an approximate joint posterior density. This is an alternative to a Metropolis-Hastings (MH) algorithm normally used to draw samples from the joint posterior density. We illustrate our method using a Poisson regression model which has much interest for the analysis of rare events from small areas. We use three examples and a simulation study to assess the performance of our method relative to the MH algorithm.