Nonparametric hierarchical models based on completely random measures

A very active line of research in Bayesian statistics has aimed at defining and investigating general classes of nonparametric priors. A notable example, which includes the Dirichlet process, is obtained through normalization or transformation of completely random measures. These have been extensively studied for the exchangeable setting. However in a large variety of applied problems data are heterogeneous, being generated by different, though related, experiments; in such situations partial exchangeability is a more appropriate assumption. In this spirit we propose a nonparametric hierarchical model based on transformations of completely random measures, which extends the hierarchical Dirichlet process. The model allows us to handle related groups of observations, creating a borrowing of strength between them. From the theoretical viewpoint, we analyze the induced partition structure, which plays a pivotal role in a very large number of inferential problems. The resulting partition probability function has a feasible expression, suitable to address predictionin its generality, as suggested by de Finetti. Finally we propose a set of applications which include inference on genomic and survival data.