A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics

TL;DR

This paper introduces a categorical framework for probability and statistics, unifying various concepts like conditional independence and sufficient statistics across different probability theories.

Contribution

It develops Markov categories as an abstract, uniform framework for probability and statistical theorems, enhancing conceptual clarity and broad applicability.

Findings

Unified categorical treatment of conditioning and disintegration

Formalization of theorems on sufficient statistics in a categorical setting

Applicability to diverse probability theories including measure-theoretic and Gaussian

Abstract

We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; various versions of conditional independence and its standard properties; conditional products; almost surely; sufficient statistics; versions of theorems on sufficient statistics due to Fisher--Neyman, Basu, and Bahadur. Besides the conceptual clarity offered by our categorical setup, its main advantage is that it provides a uniform treatment of various types of probability theory, including discrete probability theory, measure-theoretic probability with general measurable spaces, Gaussian probability, stochastic processes of either of these kinds, and many others.