Compositionality

The open-access journal for the mathematics of composition

Infinite products and zero-one laws in categorical probability

Tobias Fritz1 and Eigil Fjeldgren Rischel2

1Deparment of Mathematical Sciences, University of Copenhagen, 2100 Denmark
2Perimeter Institute for Theoretical Physics, N2L 2Y5, Waterloo, Ontario, Canada

Updated version: The authors have uploaded version v6 of this work to the arXiv which may contain updates or corrections not contained in the published version v5. The authors left the following comment on the arXiv:
20 pages. v6: Some minor fixes

ABSTRACT

Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products $\bigotimes_{i \in J} X_i$ come in two versions: a weaker but more general one for families of objects $(X_i)_{i \in J}$ in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories.

As a first application, we state and prove versions of the zero--one laws of Kolmogorov and Hewitt--Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.

► BibTeX data

► References

[1] Erik Sparre Andersen and Børge Jessen. On the introduction of measures in infinite product sets. Kongelige Danske Videnskabernes Selskab, Matematiske-Fysiske meddelelser, 25(4):8, 1948.

[2] Bruce E. Blackadar. Infinite tensor products of $C\sp*$-algebras. Pacific Journal of Mathematics, 72(2):313–334, 1977.

[3] Kenta Cho and Bart Jacobs. Disintegration and Bayesian inversion via string diagrams. Mathematical Structures in Computer Science, 29:938–971, 2019. https:/​/​doi.org/​10.1017/​S0960129518000488.
https:/​/​doi.org/​10.1017/​S0960129518000488

[4] Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih. Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1982. https:/​/​doi.org/​10.1007/​978-3-540-38955-2.
https:/​/​doi.org/​10.1007/​978-3-540-38955-2

[5] D. H. Fremlin. Measure theory. Vol. 4. Torres Fremlin, Colchester, 2006. Topological measure spaces. Part I, II, Corrected second printing of the 2003 original. https:/​/​www1.essex.ac.uk/​maths/​people/​fremlin/​mt.htmwww1.essex.ac.uk/​maths/​people/​fremlin/​mt.htm.
https:/​/​www1.essex.ac.uk/​maths/​people/​fremlin/​mt.htm

[6] Tobias Fritz. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. https:/​/​doi.org/​10.1016/​j.aim.2020.107239.
https:/​/​doi.org/​10.1016/​j.aim.2020.107239

[7] Tobias Fritz, Paolo Perrone, and Sharwin Rezagholi. Probability, valuations, hyperspace: Three monads on top and the support as a morphism. ArXiv, abs/​1910.03752, 2019.

[8] Malte Gerhold, Stephanie Lachs, and Michael Schürmann. Categorial Lévy processes. https:/​/​arxiv.org/​abs/​1612.05139arXiv:1612.05139.
arXiv:1612.05139

[9] Michèle Giry. A categorical approach to probability theory. In Categorical aspects of topology and analysis (Ottawa, Ont., 1980), volume 915 of Lecture Notes in Mathematics, pages 68–85. Springer, 1982. https:/​/​doi.org/​10.1007/​BFb0092872.
https:/​/​doi.org/​10.1007/​BFb0092872

[10] Peter V. Golubtsov. Axiomatic description of categories of information converters. Problemy Peredachi Informatsii, 35(3):80–98, 1999. In Russian. English translation in Problems of Information Transmission 35(3):259–274, 1999.

[11] Edwin Hewitt and Leonard J. Savage. Symmetric measures on cartesian products. Transactions of the American Mathematical Society, 80(2):470–470, February 1955. https:/​/​doi.org/​10.1090/​s0002-9947-1955-0076206-8.
https:/​/​doi.org/​10.1090/​s0002-9947-1955-0076206-8

[12] Georg Karner. Continuous monoids and semirings. Theoretical Computer Science, 318(3):355–372, 2004. https:/​/​doi.org/​10.1016/​j.tcs.2004.01.020.
https:/​/​doi.org/​10.1016/​j.tcs.2004.01.020

[13] Achim Klenke. Probability theory, a comprehensive course. Universitext. Springer, London, second edition, 2014. https:/​/​doi.org/​10.1007/​978-1-4471-5361-0.
https:/​/​doi.org/​10.1007/​978-1-4471-5361-0

[14] A. N. Kolmogorov. Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin, 1933.

[15] D. Ramachandran. Perfect mixtures of perfect measures. Annals of Probability, 7(3):444–452, 1979. https:/​/​doi.org/​10.1214/​aop/​1176995045.
https:/​/​doi.org/​10.1214/​aop/​1176995045

[16] Andrea Schalk. Algebras for generalized power constructions. PhD thesis, University of Darmstadt, 1993. Available at www.cs.man.ac.uk/​ schalk/​publ/​diss.ps.gzwww.cs.man.ac.uk/​ schalk/​publ/​diss.ps.gz.
http:/​/​www.cs.man.ac.uk/​~schalk/​publ/​diss.ps.gz

Cited by

[1] Tobias Fritz and Wendong Liang, "Free gs-Monoidal Categories and Free Markov Categories", Applied Categorical Structures 31 2, 21 (2023).

[2] Dario Stein and Sam Staton, 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1 (2021) ISBN:978-1-6654-4895-6.

[3] Paolo Perrone, "Markov Categories and Entropy", IEEE Transactions on Information Theory 70 3, 1671 (2024).

[4] Swaraj Dash, Younesse Kaddar, Hugo Paquet, and Sam Staton, "Affine Monads and Lazy Structures for Bayesian Programming", Proceedings of the ACM on Programming Languages 7 POPL, 1338 (2023).

[5] Elena Di Lavore and Mario Román, 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1 (2023) ISBN:979-8-3503-3587-3.

[6] Sean Moss and Paolo Perrone, Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science 1 (2022) ISBN:9781450393515.

[7] Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, Antonio Lorenzin, Paolo Perrone, and Dario Stein, "Dilations and information flow axioms in categorical probability", Mathematical Structures in Computer Science 33 10, 913 (2023).

[8] Dario Stein and Sam Staton, "Probabilistic Programming with Exact Conditions", Journal of the ACM 71 1, 1 (2024).

[9] SEAN MOSS and PAOLO PERRONE, "A category-theoretic proof of the ergodic decomposition theorem", Ergodic Theory and Dynamical Systems 43 12, 4166 (2023).

[10] Luca Giorgetti, Arthur J Parzygnat, Alessio Ranallo, and  Benjamin P Russo, "Bayesian inversion and the Tomita–Takesaki modular group", The Quarterly Journal of Mathematics 74 3, 975 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-28 07:08:32). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2024-03-28 07:08:32).