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November, 1974 Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems
Charles E. Antoniak
Ann. Statist. 2(6): 1152-1174 (November, 1974). DOI: 10.1214/aos/1176342871


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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Charles E. Antoniak. "Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems." Ann. Statist. 2 (6) 1152 - 1174, November, 1974.


Published: November, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0335.60034
MathSciNet: MR365969
Digital Object Identifier: 10.1214/aos/1176342871

Primary: 60K99
Secondary: 60G35 , 62C10 , 62G99

Keywords: Bayes , bio-assay , Dirichlet process , discrimination , Empirical Bayes , mixing distribution , nonparametric , Random measures

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 6 • November, 1974
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