1. Prior Processes

This chapter is devoted to introducing various prior processes, their formulation, properties, inter-relationships, and their relative strengths and weaknesses. The sequencing of presentation of these priors reflects mostly the order in which they were discovered and developed. The Dirichlet process and its immediate generalizations—Dirichlet Invariant and Mixtures of Dirichlet—are presented first. The neutral to the right processes and the processes with independent increments, which form the basis for many other processes, are discussed next. They are key in the development of processes that include beta, gamma and extended gamma processes, proposed primarily to address specific applications in the reliability theory, are presented next. Beta-Stacy process which extends the Dirichlet process is discussed thereafter. Following that, tailfree and Polya tree processes are presented which are especially convenient for estimating density functions, and to place greater weights, where it is deemed appropriate, by selecting suitable partitions in developing the prior. In order to extend the nonparametric Bayesian analysis to covariate data, numerous extensions are proposed. They have origin in the Ferguson-Sethuraman infinite sum representation in which the weights are constructed by a stick-breaking construction. They are collectively called here as Ferguson-Sethuraman processes and include dependent and spatial Dirichlet processes, Pitman-Yor process, Chinese restaurant and Indian buffet processes, etc. They all are included in this chapter.