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Journal Article

Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems

Charles E. Antoniak
The Annals of Statistics
Vol. 2, No. 6 (Nov., 1974), pp. 1152-1174
Stable URL: http://www.jstor.org/stable/2958336
Page Count: 23
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Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems
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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.

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