Hidden Markov Models and Disease Mapping

We present new methodology to extend hidden Markov models to the spatial domain, and use this class of models to analyze spatial heterogeneity of count data on a rare phenomenon. This situation occurs commonly in many domains of application, particularly in disease mapping. We assume that the counts follow a Poisson model at the lowest level of the hierarchy, and introduce a finite-mixture model for the Poisson rates at the next level. The novelty lies in the model for allocation to the mixture components, which follows a spatially correlated process, the Potts model, and in treating the number of components of the spatial mixture as unknown. Inference is performed in a Bayesian framework using reversible jump Markov chain Monte Carlo. The model introduced can be viewed as a Bayesian semiparametric approach to specifying flexible spatial distribution in hierarchical models. Performance of the model and comparison with an alternative well-known Markov random field specification for the Poisson rates are demonstrated on synthetic datasets. We show that our allocation model avoids the problem of oversmoothing in cases where the underlying rates exhibit discontinuities, while giving equally good results in cases of smooth gradient-like or highly autocorrelated rates. The methodology is illustrated on an epidemiologic application to data on a rare cancer in France.