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.
Highlights
A variety of ways exists to construct the Dirichlet process. For this particular case of a random probability measure, the spectrum of construction approaches ranges from the projective limit construction from finite-dimensional Dirichlet distributions proposed by Ferguson [8] to the stick-breaking construction of Sethuraman [25]; see e.g. the survey by Walker et al [27] for an overview
The purpose of this paper is to provide a projective limit result directly applicable to the construction of any probability distribution on M(V )
We address problem (i) by means of a generalization of Kolmogorov’s extension theorem, due to Bochner [4]; problem (ii) by means of the fact that the Borel σ-algebra of a Polish space V is generated by a countable subsystem of sets, which allows us to substitute the uncountable-dimensional projective limit space by a countable-dimensional surrogate; and problem (iii) using a result of Harris [14] on σ-additivity of set functions on Polish spaces
Summary
A variety of ways exists to construct the Dirichlet process For this particular case of a random probability measure, the spectrum of construction approaches ranges from the projective limit construction from finite-dimensional Dirichlet distributions proposed by Ferguson [8] to the stick-breaking construction of Sethuraman [25]; see e.g. the survey by Walker et al [27] for an overview. Most of these constructions are bespoke representations more or less specific to the Dirichlet. Appendix A reviews problems (i)-(iii) above in more detail
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