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Probability Sheaves and the Giry Monad

Author Alex Simpson

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Keywords
  • Random variable
  • conditional independence
  • category theory
  • sheaves
  • Giry monad

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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.

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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

Author Details

Alex Simpson

References

  1. M. Giry. A categorical approach to probability theory. In Categorical Aspects of Topology and Analysis, pages 68-85. Springer-Verlag, 1982. Google Scholar
  2. P. T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press, 2002. Google Scholar
  3. A. Pitts. Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press, 2013. Google Scholar
  4. A. Simpson. Category-theoretic structure for independence and conditional independence. In Proceedings of MFPS, 2017. Google Scholar
  5. 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/.
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