Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems
The Annals of Statistics · 1974 · Volume 2 · Issue 6
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.
Keywords
60G35, 60K99, 62C10, 62G99, Bayes, bio-assay, Dirichlet process, discrimination, Empirical Bayes, mixing distribution, nonparametric, Random measures
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