8 - More nonparametric Bayesian models for biostatistics

Summary

In this companion to Chapter 7 we discuss and extend some of the models and inference approaches introduced there. We elaborate on the discussion of random partition priors implied by the Dirichlet process. We review some additional variations of dependent Dirichlet process models and we review in more detail the Pólya tree prior used briefly in Chapter 7. Finally, we review variations of Dirichlet process models for data formats beyond continuous responses.

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.