Stochastic maps - 03 - Products of conditional probabilities and a.e. equivalence
Channel: Arthur Parzygnat
Date: 2019-12-28

Just as we can take the Cartesian product of functions, we can define the Cartesian product of stochastic maps by multiplying probabilities. This turns the category of stochastic maps into a monoidal category (warning: it is not Cartesian). It is the deep insight of Cho and Jacobs that the notion of almost everywhere equivalence has a diagrammatic formulation, which we review here. This idea is used in the uniqueness of Bayesian inference, which will be described in the next video. Cho and Jacob's work "Disintegration and Bayesian inversion via string diagrams" is freely available at https://arxiv.org/abs/1709.00322.

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 video is also part of 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 next video in this series is on Bayes' theorem: https://youtu.be/AAt05h1PEEQ

The previous video in this series is on Composing conditional probabilities: https://youtu.be/rsKOFwClQpU

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