Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems

Antoniak, Charles E.

Ann. Statist., Tome 2 (1974) no. 1, p. 1152-1174 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1974-11-14
Classification:  Dirichlet process,  nonparametric,  Bayes,  empirical Bayes,  mixing distribution,  random measures,  bio-assay,  discrimination,  60K99,  60G35,  62C10,  62G99