This is the last of my three talks on Markov categories, and in this talk, we are going to use our categorical knowledge of Markov categories to prove one of the most famous theorems in probability theory: the De Finetti theorem, which states that a collection of exchangeable random variables is equivalent to a mixture of independent and identically distributed random variables. To prove the De Finetti theorem using Markov categories, we will need to define notions like almost sure equality, conditional distributions, and deterministic maps purely categorically and show that they correspond to the usual notions when realized in the category BorelStoch of Polish spaces and stochastic processes between Polish spaces. Once we do this, we will be able to talk about Markov categories with conditionals, Markov categories with almost sure compatible representability, and Markov categories with Kolmogorov Powers. Then, we prove using monoidal graphical calculus an abstract version of the De Finetti theorem for Markov categories that have these three properties, which reduces to the ordinary De Finetti theorem in the setting of BorelStoch.
Graduate Student Geometry-Topology Seminar
Friday, May 9, 2025 - 2:00pm
Isaiah Hilsenrath
University of Pennsylvania
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