Nonparametric Statistical Methods, 3rd Edition

Chapter 16

An Introduction to Bayesian Nonparametric Statistics via the Dirichlet Process

Introduction

Bayesian statistics incorporate prior information about the parameter of interest into the inferential method. The prior information is specified through the use of a prior distribution on the parameter of interest, thereby treating that parameter as a random quantity. The prior information can be obtained in many ways including pilot experiments and the opinions of experts. The parameter is chosen by the prior and then after the data are obtained, the posterior distribution is computed. The posterior distribution is the conditional distribution of the parameter, given the data. If more data are obtained, the posterior is used as the new prior and then a new posterior distribution is computed. This is called Bayesian updating.

It is easiest to do Bayesian statistics when the unknown parameter lies in a finite-dimensional space. For example, in the problem of estimating a success probability considered in Chapter 2, the typical prior used for the one-dimensional parameter is a member of the Beta family whose density function is proportional to . Then after observing the outcomes of ...