### A Categorical View of Conditional Expectation

#### Posted by John Baez

I always like to see categories combined with probability theory and analysis. So I’m glad Prakash Panangaden did that in his talk at the ACT@UCR seminar. Afterwards we had discussions at the Category Theory Community Server, and you can see them here if you’re a member:

https://categorytheory.zulipchat.com/

You can see his slides here, or download a video here, or watch the video here.

Here’s what his talk is about:

- Prakash Panangaden: A categorical view of conditional expectation.

Abstract.This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones—which are abstract versions of the familiar cones in vector spaces—of measures and related categories cones of L_{p}functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

For more, see:

- Philippe Chaput, Vincent Danos, Prakash Panangaden and Gordon Plotkin, Approximating Markov processes by averaging, in
*International Colloquium on Automata, Languages, and Programming*, Springer, Berlin, 2009.

## Re: A Categorical View of Conditional Expectation

You can now see Panangaden’s slides here, or download a video here, or watch the video here.