Stochastic maps - 04 -Bayes' theorem
Channel: Arthur Parzygnat
Date: 2019-12-28

We finally state Bayes' theorem in a categorical language. This version of the theorem states that given a measure-preserving stochastic map, there exists a measure-preserving stochastic map in the opposite direction that satisfies a diagrammatic condition equivalent to Bayes' rule. This Bayesian inverse is unique almost surely. 

The subject of categorical probability is an area of active research. Here are some links to papers on the subject, many of which discuss the theorem presented here in some form.
Culbertson and Sturtz - A categorical foundation for Bayesian probability https://arxiv.org/abs/1205.1488
Fong - Causal Theories: A Categorical Perspective on Bayesian Networks https://arxiv.org/abs/1301.6201
Cho and Jacobs - Disintegration and Bayesian Inversion via String Diagrams https://arxiv.org/abs/1709.00322
Fritz - A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics https://arxiv.org/abs/1908.07021

This is part of a series of lectures on special topics in linear algebra https://www.youtube.com/playlist?list=PLSx1kJDjrLRSS4Aio9WfQlNnH6X6pC_8t. It is assumed the viewer has taken (or is well into) a course in linear algebra. Some topics may also require additional background. Topics covered include linear regression and data analysis, ordinary linear differential equations, differential operators, function spaces, category-theoretic aspects of linear algebra, support vector machines (machine learning), Hamming's error-correcting codes, stochastic maps and Markov chains, tensor products, finite-dimensional C*-algebras, algebraic probability theory, completely positive maps, aspects of quantum information theory, and more.

This is the last video from the playlist on Categorical probability theory https://www.youtube.com/playlist?list=PLSx1kJDjrLRSKKHj4zetTZ45pVnGCRN80
and also on Bayes' theorem https://www.youtube.com/playlist?list=PLSx1kJDjrLRQksb7H9fqRE8GVMJdkX-4A

The previous video in this series is on Products of conditional probabilities and a.e. equivalence: https://youtu.be/B9InTjCb1GQ

The next video on Special Topics in Linear Algebra is on *-homomorphisms of finite-dimensional C*-algebras: https://youtu.be/NbgpY0hkFi0

These videos were created during the 2019 Spring/Summer semester at the UConn CETL Lightboard Room.