Bayesian analysis of Poisson mixtures

Valerie Viallefont, Sylvia Richardson & Peter J. Green
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 of the mixture (number of components, weights, and Poisson parameters) will be estimated by MCMC algorithms, including the reversible jump algorithm which permits varying the dimension between two states of the chain. We will focus on the difficulty of finding a weakly informative prior for the Poisson parameters distribution: different priors will be detailed and compared.

Then the mixing performances of different moves created for dimension change will be investigated. The model is extended by the introduction of covariates, whose effect can be either uniform or different among the components. Simulated data sets will be designed for the different comparisons, and the model will finally be illustrated on real data.

Some key words: Reversible jump, Heterogeneity, Markov chain Monte Carlo, Poisson mixtures, Priors
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