Disintegration and Bayesian Inversion via String Diagrams

TL;DR

This paper presents an abstract graphical framework for disintegration and Bayesian inversion in probability theory, providing foundational results and illustrating their applications across different probability spaces.

Contribution

It introduces a novel abstract, diagrammatic approach to disintegration and Bayesian inversion, unifying their treatment in measure-theoretic and discrete probability contexts.

Findings

Disintegration and Bayesian inversion are formalized using string diagrams.

Existence results are established for discrete and measure-theoretic probability.

Applications demonstrate the practical utility of the abstract formulations.

Abstract

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.