Bayesian analysis of Poisson mixtures

The modelling of rare events via a Poisson distribution sometimes reveals substantial over-dispersion, indicating that some unexplained discontinuity arises in the data. We suggest modelling this over-dispersion by a Poisson mixture. In a hierarchical Bayesian model, the posterior distributions of the unknown quantities in the mixture (number of components, weights, and Poisson parameters) will be estimated by MCMC algorithms, including reversible jump algorithms which permits varying the dimension of the mixture. We will focus on the difficulty of finding a weakly informative prior for the Poisson parameters: different priors will be detailed and compared. Then, the performances of different moves created for changing dimension will be investigated. The model is extended by the introduction of covariates, with homogeneous or heterogeneous effect. Simulated data sets will be designed for the different comparisons, and the model will finally be illustrated on real data.