Absolute continuity, supports and idempotent splitting in categorical probability
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915791748726784 |
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| author | Fritz, Tobias Gonda, Tomáš Lorenzin, Antonio Perrone, Paolo Stein, Dario |
| author_facet | Fritz, Tobias Gonda, Tomáš Lorenzin, Antonio Perrone, Paolo Stein, Dario |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00651 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Absolute continuity, supports and idempotent splitting in categorical probability Fritz, Tobias Gonda, Tomáš Lorenzin, Antonio Perrone, Paolo Stein, Dario Probability Logic in Computer Science Category Theory 18M05, 60A05 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. |
| title | Absolute continuity, supports and idempotent splitting in categorical probability |
| topic | Probability Logic in Computer Science Category Theory 18M05, 60A05 |
| url | https://arxiv.org/abs/2308.00651 |