Does a Gibbs sampler approach to spatial Poisson regression models outperform a single site MH sampler?

A Gibbs sampler for a Poisson regression model including spatial effects is presented and evaluated. The approach is based on that a Poisson regression model can be transformed into an approximate normal linear model by data augmentation using the introduction of two sequences of latent variables. It is shown how this methodology can be extended to spatial Poisson regression models and details of the resulting Gibbs sampler are given. In particular, the influence of model parameterisation and different update strategies on the mixing of the MCMC chains is discussed. The developed Gibbs samplers are analysed in two simulation studies and applied to model the expected number of claims for policyholders of a German car insurance company. The mixing of the Gibbs samplers depends crucially on the model parameterisation and the update schemes. The best mixing is achieved when collapsed algorithms are used, reasonable low autocorrelations for the spatial effects are obtained in this case. For the regression effects however, autocorrelations are rather high, especially for data with very low heterogeneity. For comparison a single component Metropolis-Hastings algorithms is applied which displays very good mixing for all components. Although the Metropolis-Hastings sampler requires a higher computational effort, it outperforms the Gibbs samplers which would have to be run considerably longer in order to obtain the same precision of the parameters.