Electronic Journal of Statistics

Projective limit random probabilities on Polish spaces

Peter Orbanz

Full-text: Open access

Abstract

A pivotal problem in Bayesian nonparametrics is the construction of prior distributions on the space M(V) of probability measures on a given domain V. In principle, such distributions on the infinite-dimensional space M(V) can be constructed from their finite-dimensional marginals—the most prominent example being the construction of the Dirichlet process from finite-dimensional Dirichlet distributions. This approach is both intuitive and applicable to the construction of arbitrary distributions on M(V), but also hamstrung by a number of technical difficulties. We show how these difficulties can be resolved if the domain V is a Polish topological space, and give a representation theorem directly applicable to the construction of any probability distribution on M(V) whose first moment measure is well-defined. The proof draws on a projective limit theorem of Bochner, and on properties of set functions on Polish spaces to establish countable additivity of the resulting random probabilities.

Article information

Source
Electron. J. Statist. Volume 5 (2011), 1354-1373.

Dates
First available in Project Euclid: 19 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1319028571

Digital Object Identifier
doi:10.1214/11-EJS641

Mathematical Reviews number (MathSciNet)
MR2842908

Zentralblatt MATH identifier
1274.62076

Subjects
Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 60G57: Random measures

Keywords
Bayesian nonparametrics Dirichlet processes random probability measures

Citation

Orbanz, Peter. Projective limit random probabilities on Polish spaces. Electron. J. Statist. 5 (2011), 1354--1373. doi:10.1214/11-EJS641. https://projecteuclid.org/euclid.ejs/1319028571.


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