π€ AI Summary
This paper addresses the categorical decomposition of probabilistic structures in Markov categories, establishing a rigorous categorical foundation for absolute continuity, support sets, and idempotent splittings. Methodologically, it introduces, for the first time, an idempotent splitting theorem for measurable Markov kernels within the category of standard Borel spaces, and distills a general splitting criterion applicable to arbitrary Markov categories. The main contributions are: (1) a precise internal categorical definition of support sets; (2) a proof that every idempotent measurable Markov kernel between standard Borel spaces admits a splitting; and (3) a rigorous, broadly applicable theoretical framework for structural decomposition of probabilistic models, categorical modeling of stochastic processes, and abstract Bayesian inference.
π Abstract
Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like conditional probability and almost sure equality, as well as proofs of fundamental results such as the Hewitt-Savage 0/1 Law, the de Finetti Theorem and the Ergodic Decomposition Theorem. In this work, we develop additional relevant notions from probability theory in the setting of Markov categories. This comprises improved versions of previously introduced definitions of absolute continuity and supports, as well as a detailed study of idempotents and idempotent splitting in Markov categories. Our main result on idempotent splitting is that every idempotent measurable Markov kernel between standard Borel spaces splits through another standard Borel space, and we derive this as an instance of a general categorical criterion for idempotent splitting in Markov categories.
