Projective Limit Random Probabilities on Polish Spaces

TL;DR

This paper develops a method for constructing prior distributions on the space of probability measures over Polish spaces using projective limits, addressing technical challenges in infinite-dimensional Bayesian nonparametrics.

Contribution

It provides a representation theorem for constructing probability distributions on M(V) with well-defined first moments, leveraging projective limit theorems and properties of Polish spaces.

Findings

Resolved technical difficulties in constructing priors on M(V) for Polish spaces.

Established countable additivity of the constructed random probabilities.

Provided a direct representation theorem applicable to various distributions.

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.