Disintegration and Bayesian Inversion via String Diagrams
This work provides a foundational abstraction for probability theory, which is incremental as it reformulates existing concepts in a new graphical framework.
The paper tackles the problem of formalizing disintegration and Bayesian inversion in conditional probability theory by presenting them in abstract graphical formulations, and it demonstrates their existence for discrete and measure-theoretic probability while illustrating their usefulness in examples.
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.