Eur. Phys. J. Plus (2021) 136: 441
https://doi.org/10.1140/epjp/s13360-021-01427-7

Regular Article

Probabilistic morphisms and Bayesian nonparametrics

¹ Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany
² Institute of Mathematics of the Czech Academey of Sciences, Zitna 25, 11567, Praha 1, Czech Republic
³ Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109, Leipzig, Germany

^b hvle@math.cas.cz

Received: 29 September 2020
Accepted: 10 April 2021
Published online: 24 April 2021

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.