A categorical foundation for Bayesian probability

TL;DR

This paper develops a categorical framework for Bayesian probability, establishing the existence and uniqueness of inference maps in a general setting that includes decision rules and iterative updating.

Contribution

It introduces a categorical foundation for Bayesian probability that extends to non-Polish spaces and incorporates decision rules within a unified framework.

Findings

Existence of inference maps in a broad measurable space setting.

Framework supports iterative Bayesian updating.

Includes decision rules in the categorical Bayesian framework.

Abstract

Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu: 1 \rightarrow D$, a posterior probability $\widehat{P_H}= I \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $I$ and the process repeats with each additional measurement. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the setting of Polish spaces. This less stringent setting then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.