Categorical algebra of conditional probability

TL;DR

This paper explores the categorical algebra underlying conditional probability, establishing connections between Markov categories, probability monads, and statistical experiments, with a focus on the Giry monad and universal properties.

Contribution

It introduces a categorical framework linking weakly cartesian functors and natural transformations to conditioning, extending the understanding of probability structures in category theory.

Findings

Giry monad on standard Borel spaces satisfies Beck-Chevalley condition

Standard measure construction has a universal property for deterministic experiments

Connects categorical algebra to statistical experiment theory

Abstract

In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and statistics. In this paper, we connect some ideas from categorical algebra, namely weakly cartesian functors and natural transformations, to the idea of conditioning in probability theory, using Markov categories and probability monads. First of all, we show that under some conditions, the monad associated to a Markov category with conditionals has a weakly cartesian functor and weakly cartesian multiplication (a condition known as Beck-Chevalley, or BC). In particular, we show that this is the case for the Giry monad on standard Borel spaces. We then connect this theory to existing results on statistical experiments. We show that for deterministic statistical experiments, the so-called standard measure construction (which can be seen as a generalization of the "hyper-normalizations" introduced by Jacobs) satisfies a universal property, allowing an equivalent definition which does not rely on the existence of conditionals.