Stochastic maps - 01 - Conditional probabilities
Channel: Arthur Parzygnat
Date: 2019-12-28

Stochastic maps, also called stochastic matrices, conditional probabilities, and Markov kernels, are probabilistic generalizations of functions that associate probability measures to points. This idea is crucial in the formulation of Bayes' theorem, whose statement is a goal of these few videos. I learned about stochastic maps from Baez and Fritz's work "A Bayesian Characterization of Relative Entropy" http://www.tac.mta.ca/tac/volumes/29/16/29-16abs.html. 

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 Composing conditional probabilities: https://youtu.be/rsKOFwClQpU

The previous video in this series is on Markov kernels: https://youtu.be/psUDrasc21o (but it is not needed to understand this video!)

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