Efficient and accurate approximate Bayesian inference with an application to insurance data

We propose an efficient and accurate approximate Bayesian Markov chain Monte Carlo methodology for the estimation of event rates under an overdispersed Poisson distribution. A Gibbs sampling algorithm is derived, based on a log-normal/gamma mixture density that closely approximates the conditional distribution of the Poisson parameters. This involves a moment matching process, with the exact conditional moments obtained employing an entropy distance minimisation (KullbackLiebler divergence) criterion. A simulation study is conducted and demonstrates good Bayes risk properties and robust performance for the estimator, as compared with other estimating approaches under various loss functions. Actuarial data on insurance claims are used to illustrate the methodology. The approximate analysis displays superior Markov chain Monte Carlo convergence in terms of effective sample size, whilst providing almost identical inferences to those obtained with an exact method. keywords: Bayes risk; Entropy distance; Effective sample size; Gibbs sampling; Hierarchical Bayesian analysis; Insurance claims; Markov chain Monte Carlo; Mixture distribution; Overdispersion