LIPIcs.CALCO.2017.1.pdf
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Probability Sheaves and the Giry Monad
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7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Conference on Algebra and Coalgebra in Computer Science (CALCO) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2017-11-17
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Alex Simpson. Probability Sheaves and the Giry Monad. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 1:1-1:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CALCO.2017.1
Abstract
I introduce the notion of probability sheaf, which is a mathematical structure capturing the relationship between probabilistic concepts (such as random variable) and sample spaces. Various probability-theoretic notions can be (re)formulated in terms of category-theoretic structure on the category of probability sheaves. As a main example, I consider the Giry monad, which, in its original formulation, constructs spaces of probability measures. I show that the Giry monad generalises to the category of probability sheaves, where it turns out to have a simple, purely category-theoretic definition.
Subject Classification
Keywords
- Random variable
- conditional independence
- category theory
- sheaves
- Giry monad
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References
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M. Giry. A categorical approach to probability theory. In Categorical Aspects of Topology and Analysis, pages 68-85. Springer-Verlag, 1982.
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P. T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press, 2002.
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A. Pitts. Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press, 2013.
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A. Simpson. Category-theoretic structure for independence and conditional independence. In Proceedings of MFPS, 2017.
- T. Tao. A review of probability theory. Note0 in 254A - random matrices, 2010. URL: https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/.