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

Efficient and accurate Bayesian Markov chain Monte Carlo methodology is proposed for the estimation of event rates under an overdispersed Poisson distribution. An approximate Gibbs sampling method and an exact independence-type Metropolis–Hastings algorithm are 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 (Kullback–Liebler divergence) criterion. A simulation study is conducted and demonstrates good Bayes risk properties and robust performance for the proposed estimators, 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 mixing efficiency, whilst providing almost identical inferences to those obtained with exact methods.