# Probabilistic morphisms and Bayesian nonparametrics

## TL;DR

This paper develops a functorial framework for probabilistic morphisms and applies it to Bayesian nonparametrics, providing new insights into posterior distributions and Dirichlet measures within a categorical setting.

## Contribution

It unifies existing categories of probabilistic morphisms and statistical models, introduces a formal Bayesian model with explicit posterior formulas, and offers a new proof for Dirichlet measures using functorial properties.

## Key findings

- Explicit formula for posterior distributions in Bayesian models with specific conditions.
- Unified categorical framework for probabilistic morphisms and statistical models.
- New proof of the existence of Dirichlet measures using functorial properties.

## 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.