Spatial heterogeneity plays an important role in a diverse set of applications. For example, in ecology, heterogeneous environments promote camouflaged prey species and disruptive selection; in economics, local characteristics determine regional policies; in health sciences, histopathological heterogeneity characterizes certain cancer tissues within and among tumor types.

Recent Bayesian modeling of univariate spatial data has considered mixed effect models, where a residual stationary (homogeneous) Gaussian effect is assumed. Arguably, one might prefer the flexibility of a nonstationary, non-Gaussian specification. In a nonparametric setting, this can be accommodated by mixture of Dirichlet process (DP) models. The DP is an example of a species sampling prior, which are typically used to describe diversity of different ecological groups of species under different environmental conditions.

However, a limitation of the mixture of DP models is that the latent factor driving species sampling is globally defined and may fail to account for spatial heterogeneity. In this work, we introduce a novel class of prior distributions, the hybrid Dirichlet Processes (hDP), which generalize the DP and overcome this limitation. In a spatial setting, the hDP are defined as mixtures of Gaussian random fields with spatially varying weights. A crucial feature of this specification is the possibility to model local speciation and hybrid clustering.

We illustrate the procedure by means of a simulated example and an application to the analysis of hippocampal atrophy in brains of patients affected by Alzheimer's disease.

This is joint work with Alan Gelfand (Duke University, USA) and Sonia Petrone (Bocconi University, Italy).