## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 6 (1974), 1152-1174.

### Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems

#### Abstract

A random process called the Dirichlet process whose sample functions are almost surely probability measures has been proposed by Ferguson as an approach to analyzing nonparametric problems from a Bayesian viewpoint. An important result obtained by Ferguson in this approach is that if observations are made on a random variable whose distribution is a random sample function of a Dirichlet process, then the conditional distribution of the random measure can be easily calculated, and is again a Dirichlet process. This paper extends Ferguson's result to cases where the random measure is a mixing distribution for a parameter which determines the distribution from which observations are made. The conditional distribution of the random measure, given the observations, is no longer that of a simple Dirichlet process, but can be described as being a mixture of Dirichlet processes. This paper gives a formal definition for these mixtures and develops several theorems about their properties, the most important of which is a closure property for such mixtures. Formulas for computing the conditional distribution are derived and applications to problems in bio-assay, discrimination, regression, and mixing distributions are given.

#### Article information

**Source**

Ann. Statist. Volume 2, Number 6 (1974), 1152-1174.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176342871

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aos/1176342871

**Mathematical Reviews number (MathSciNet)**

MR365969

**Zentralblatt MATH identifier**

0335.60034

**Subjects**

Primary: 60K99: None of the above, but in this section

Secondary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 62C10: Bayesian problems; characterization of Bayes procedures 62G99: None of the above, but in this section

**Keywords**

Dirichlet process nonparametric Bayes empirical Bayes mixing distribution random measures bio-assay discrimination

#### Citation

Antoniak, Charles E. Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems. The Annals of Statistics 2 (1974), no. 6, 1152--1174. doi:10.1214/aos/1176342871. http://projecteuclid.org/euclid.aos/1176342871.