One setting in which mixtures have yet to make much impact is that of data indexed spatially, whether by points, by regions, or by cells of a lattice. In this paper, we confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in a mixture model that vary from one location to another.
Our specific focus is on Poisson distributed data, and applications in geographical epidemiology. We work in a Bayesian framework, with the Poisson parameters drawn from gamma priors, and the number of components being unknown. We propose two alternative models for spatially-dependent weights on the components, based on transformations of autoregressive gaussian processes: in one the mixture component labels are exchangeable, in the other they are ordered. The performance of both of these formulations is examined on both synthetic data, and real data on mortality from rare disease.