Introduction

In Chapter 7, Dunson introduced many interesting applications of nonparametric priors for inference in biomedical problems. The focus of the discussion was on Dirichlet process (DP) priors and variations. While the DP prior defines a probability model for a (discrete) random probability distribution G, the primary objective of inference in many recent applications is not inference on G. Instead many applications of the DP prior exploit the random partition of the Pólya urn scheme that is implied by the configuration of ties among the random draws from a discrete measure with DP prior. When the emphasis is on inference for the clustering, it is helpful to recognize the DP as a special case of more general clustering models. In particular we will review the product partition (PPM) and species sampling models (SSM). We discuss these models in Section 8.2. A definition and discussion of the SSM as a random probability measure also appears in Section 3.3.4. Another useful characterization of the DP is as a special case of the Pólya tree (PT) prior.