Categorical Probability Theory

Explore the foundations of categorical probability theory in this comprehensive 1.5-hour lecture series. Delve into Kleisli categories and their relationship with probability, starting with an introduction to the Giry monad and progressing to the Kleisli category of a monad. Examine Markov kernels and their role in probabilistic modeling. Investigate stochastic maps, focusing on conditional probabilities, their composition, and products. Learn about almost everywhere equivalence and conclude with an in-depth look at Bayes' theorem, providing a solid understanding of the mathematical underpinnings of probabilistic reasoning.

Syllabus

Kleisli categories and probability - 01 - The Giry monad.
Kleisli categories and probability - 02 - The Kleisli category of a monad.
Kleisli categories and probability - 03 - Markov kernels.
Stochastic maps - 01 - Conditional probabilities.
Stochastic maps - 02 - Composing conditional probabilities.
Stochastic maps - 03 - Products of conditional probabilities and a.e. equivalence.
Stochastic maps - 04 -Bayes' theorem.