Probabilistic morphisms and Bayesian nonparametrics

Abstract

In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms 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 revisit the existence of a posterior distribution, using probabilistic morphisms}. In particular, we give an explicit formula for posterior distributions of the Bayesian statistical model, assuming that the underlying parameter space is a Souslin space and the sample space 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.

arXiv commentFinal version: 22 p. Version 2: minor corrections, Proposition 2.10 added, improved presentation, 39 p., version 1: 38 p. comments welcome!

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Cited by in corpus (5)

• A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents
• Diffeological statistical models, the Fisher metric and probabilistic mappings
• On the asymptotic behavior of the Diaconis and Freedman's chain in a multidimensional simplex
• Nonparametric estimations and the diffeological Fisher metric