Absolute continuity, supports and idempotent splitting in categorical probability

TL;DR

This work develops a categorical treatment of probability via Markov categories, introducing refined notions of absolute continuity and supports tailored to monoidal structure and tensoring. By extending almost-sure equality with an extra input wire, it defines a robust absolute continuity preorder and its bicontinuous variant, and then builds a universal notion of supports that captures topological and measure-theoretic behavior across FinStoch, TychStoch, SetMulti, and BorelStoch. It then analyzes idempotents in Markov categories, distinguishing static, strong, and balanced types, and proves a general splitting criterion: in causal, positive, observationally representable categories satisfying the equalizer principle, every balanced idempotent splits; in particular, every idempotent Markov kernel between standard Borel spaces splits, strengthening Blackwell’s classical result. The paper also introduces the regular support completion and the Blackwell envelope to systematically adjoin supports and balance-idempotent splittings, yielding a flexible framework for statistical modeling and input-output relations within a categorical probability setting. Overall, the results illuminate how categorical structure controls fundamental probabilistic operations such as conditioning, independence, and long-run projections, with potential applications to statistics, graphical models, and ergodic theory."

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.

Figures (1)

• Figure 1: Venn diagram of types of idempotents in a Markov category.

Theorems & Definitions (244)

• Theorem
• Definition 2.1.1
• Remark 2.1.2
• Lemma 2.1.3
• proof
• Example 2.1.4
• Example 2.1.5
• Proposition 2.1.6
• proof
• Definition 2.2.1: Absolute continuity