Probabilistic morphisms and Bayesian nonparametrics

0541937 - MÚ 2022 RIV DE eng J - Journal Article
Jost, J. - Le, Hong-Van - Tran, T.D.
Probabilistic morphisms and Bayesian nonparametrics.
European Physical Journal Plus. Roč. 136, č. 4 (2021), č. článku 441. ISSN 2190-5444. E-ISSN 2190-5444
R&D Projects: GA ČR(CZ) GC18-01953J
Institutional support: RVO:67985840
Keywords : Bayesian nonparametrics * probabilistic morphisms * Kleisli category
OECD category: Pure mathematics
Impact factor: 3.758, year: 2021
Method of publishing: Limited access
https://doi.org/10.1140/epjp/s13360-021-01427-7

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.
Permanent Link: http://hdl.handle.net/11104/0319448