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Stream: theory: categorical probability

Topic: best intro in 2022

Naso (Oct 05 2022 at 01:09):

what's the best way to get up to speed in the field of categorical probability in 2022? I know about the 2020 workshop. Are there any nice & easy introductory articles, blog posts, etc. ?

Paolo Perrone (Oct 05 2022 at 13:41):

As far as I'm aware we don't have much introductory material written yet, there are some videos (some from the 2020 workshop).
What is your background in category theory and in probability?

Naso (Oct 05 2022 at 13:52):

Paolo, just watched your tutorial from the 2020 workshop earlier today -- it was awesome. The metaphor with the powerset monad was really apt and helpful.

I'm comfortable with basic category theory (up to topos theory, monoidal cats, etc.) and I once learnt some classical measure-theoretic probability theory from the book ''a first look at rigorous probability theory, 2nd ed'" by J. Rosenthal. I re-read the first chapters yesterday to refresh my memory.

Paolo Perrone (Oct 05 2022 at 14:21):

Okay, here are a few things that could help you.

Written resources:

Chapter 1 of my PhD thesis has an intuitive overview, and has most of the material that you saw in my tutorial video;
Section 2 of this paper has a little overview of the idea behind Markov categories;
The bulk of the theory of Markov categories, with many details, can be found in this seminal paper by Tobias Fritz;
Two of the most recent results in the field are a categorical proof of de Finetti's theorem and a categorical characterization and generalization of Markov properties for graphical models.

Introductory videos, besides the videos of the Categorical Probability and Statistics Workshop (link for who didn't know it), could be the following:

An improvised talk of mine about Markov categories;
A (real) talk by Tobias Fritz on Markov categories and de Finetti's theorem;
This talk by Tobias Fritz about categorical probability and the law of large numbers (work in progress).

@Tobias Fritz, @Tomáš Gonda, @Dario Stein, and others may have more material.

There is also the computer-science side of things, but I feel I'm not the right person to explain that part.

Tobias Fritz (Oct 05 2022 at 14:32):

That's great! Maybe it's worth adding pointers to the video series Kleisli categories and probability by @Arthur Parzygnat and to Probability monads, the Bachelor thesis of Julian Asilis. But other than that I also don't know any introductory material... (Both of us could certainly point to further research-level works.)

John Baez (Oct 05 2022 at 16:16):

Tom Leinster's student Ruben van Belle recently gave a talk at the Edinburgh category theory seminar that was an overview of work on categorical probability theory. His talk was on Zoom but I don't know if his talk was recorded. If you contact him he could give you his slides.

He also recommended a book by Bart Jacobs on categorical probability theory. Anyone know what it's called?

John Baez (Oct 05 2022 at 16:17):

Looking for it, I ran into these slides:

Rob Cornish, An introduction to categorical probability theory.

Their date is June 16, 2022, so they may cover some new stuff.

Ralph Sarkis (Oct 05 2022 at 16:36):

The book by Bart Jacobs is probably :laughing: this draft : http://www.cs.ru.nl/B.Jacobs/PAPERS/ProbabilisticReasoning.pdf

Tobias Fritz (Oct 05 2022 at 16:49):

Almost surely!

Matteo Capucci (he/him) (Oct 05 2022 at 21:04):

Ralph Sarkis said:

The book by Bart Jacobs is probably :laughing: this draft : http://www.cs.ru.nl/B.Jacobs/PAPERS/ProbabilisticReasoning.pdf

Wow what a treasure trove :heart_eyes:

Matteo Capucci (he/him) (Oct 05 2022 at 21:04):

I'm in love

Ruben Van Belle (Oct 06 2022 at 16:38):

John Baez said:

Tom Leinster's student Ruben van Belle recently gave a talk at the Edinburgh category theory seminar that was an overview of work on categorical probability theory. His talk was on Zoom but I don't know if his talk was recorded. If you contact him he could give you his slides.

He also recommended a book by Bart Jacobs on categorical probability theory. Anyone know what it's called?

I have uploaded the slides of my talk on my webpage (https://www.maths.ed.ac.uk/~s1945481/).

Tobias Fritz (Oct 06 2022 at 17:29):

Wow, I see that you have a brand-new paper A categorical proof of the Carathéodory extension theorem and an upcoming one on Radon-Nikodym! That looks amazing! Should give it a read one of these days.

Paolo Perrone (Nov 27 2022 at 13:52):

Here's a new video introduction to Markov categories!
https://youtu.be/uaQAoTFQebY

Xuanrui Qi (Nov 27 2022 at 15:03):

Thanks, exactly what I need!

Bradley Saul (Dec 02 2022 at 12:31):

Watched this last night; it was great. Thanks!

Paolo Perrone (Dec 23 2022 at 08:35):

Paolo Perrone said:

Here's a new video introduction to Markov categories!
https://youtu.be/uaQAoTFQebY

By the way, when I gave that talk (live) some people were asking me about metric enrichment, and about entropy. I've finally figured out the way to combine them, here it is! https://arxiv.org/abs/2212.11719
(There's also introductory material in this preprint, from the point of view of information theory.)

Rob Cornish (Dec 23 2022 at 12:39):

Looks extremely cool!