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Projective limit random probabilities on Polish spaces. (English) Zbl 1274.62076
Summary: 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.

##### MSC:
 62C10 Bayesian problems; characterization of Bayes procedures 62G99 Nonparametric inference 60G57 Random measures
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