Stochastic maps - 02 - Composing conditional probabilities
Channel: Arthur Parzygnat
Date: 2019-12-28

Stochastic maps can be composed and form a category. The composition is given by the Chapman--Kolmogorov equation. This composition rule reproduces the notion of push-forward of a measure (also called the law of a random variable). We will also see how the composition rule is used in the formulation of Bayes' theorem in a later video. The idea of visualizing a measure-preserving function in terms of water droplets came from Gromov's lectures on "Probability, symmetry, linearity," which are available on youtube.

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 Products of conditional probabilities and a.e. equivalence: https://youtu.be/B9InTjCb1GQ

The previous video in this series is on Conditional probabilities: https://youtu.be/xhDCmyDBj24

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