Probabilistic Mappings and Bayesian Nonparametrics
Jürgen Jost, Hông Vân Lê, Hoang Duc Luu, and Tat Dat Tran
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Submission date: 26. Jun. 2019
MSC-Numbers: 62G99, 18C20, 28C20
Keywords and phrases: statistical model, Kleisli category of probabilistic mappigs, Bayes' formula, posterior distribution, Dirichlet measure, complete Riemannian manifold
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In this paper we develop a functorial language of probabilistic mappings and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic mappings proposed by Lawvere and Giry with the category of statistical models proposed by Chentsov and Morse-Sacksteder. Then we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a given prior distribution in Bayesian statistics. We give a formula for posterior distributions, assuming that the underlying parameter space of a Bayesian statistical model is a Souslin space and the sample space of the Bayesian statistical model is a subset in a complete connected ﬁnite dimensional Riemannian manifold. Then we give a new proof of the existence of Dirichlet measures over any measurable space using a functorial property of the Dirichlet map constructed by Sethuraman.