Kleisli categories and probability - 01 - The Giry monad

This video introduces the Giry monad. A monad on a category consists of three data. In this case, it gives an endo-functor on the category of measurable spaces and measurable maps. The functor takes a measurable space to the space of probability measures and it takes a map to the pushforward of measures. In addition, there is a natural transformation that computes the barycenter of a probability measure on the space of probability measures. Finally, there is a natural transformation that views each point of a space as the Dirac delta distribution on that point.

Note: I forgot to mention that the second pair of diagrams in the definition of a monad both equal the identity natural transformation for T.

Brief history: In 1962, Bill Lawvere introduced the category of probabilistic maps. In 1982, Michele Giry viewed Lawvere's construction as the Kleisli category of a particular monad.

This video is part of the playlist on Categorical probability theory:    • Categorical probability theory  

The next video in this series is on the Kleisli category of a monad:    • Kleisli categories and probability - 02 - ...  

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