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Stream: theory: applied category theory

Topic: probability and statistics

Arthur Parzygnat (Apr 26 2020 at 20:38):

I'm opening up the topic for probability and statistics. There will no doubt be some overlap with other topics already up, but I hope this will be useful for some.

Oliver Shetler (Apr 26 2020 at 20:43):

Arthur Parzygnat: I'm opening up the topic for probability and statistics. There will no doubt be some overlap with other topics already up, but I hope this will be useful for some.

Thanks! I guess anybody can make one. Good to know!

Tobias Fritz (Apr 26 2020 at 23:36):

Thanks, Arthur, for starting this!

Those hanging out here may also be interested in the online workshop Categorical Probability and Statistics, taking place June 5-8. Everyone is welcome to attend and discuss with us!

If you haven't already done so, please let me know in case that you'd like to attend, since we will be sending around an email with the links to the sessions before it starts. The livestream on youtube will also be accessible without that. We're currently working on the schedule.

Oliver Shetler (Apr 27 2020 at 01:06):

If you haven't already done so, please let me know in case that you'd like to attend, since we will be sending around an email with the links to the sessions before it starts.

@Tobias Fritz, Please put me on the list. My email is oliver@olivershetler.com

John Baez (Apr 27 2020 at 03:54):

Yes, anyone can make one.

Tobias Fritz (Apr 27 2020 at 13:31):

Oliver Shetler said:

If you haven't already done so, please let me know in case that you'd like to attend, since we will be sending around an email with the links to the sessions before it starts.

Tobias Fritz, Please put me on the list. My email is oliver@olivershetler.com

Done. Looking forward to meeting you!

Aleks Kissinger (Apr 27 2020 at 14:34):

Hey @Tobias Fritz, could you put me on the list as well? I'm now co-organising QPL (which is going online as well), on June 2-6, so would def be interested in try to come to the later part, now that the events aren't (physically) on two different continents anymore.

Arthur Parzygnat (Apr 27 2020 at 14:46):

Aleks Kissinger said:

Hey Tobias Fritz, could you put me on the list as well? I'm now co-organising QPL (which is going online as well), on June 2-6, so would def be interested in try to come to the later part, now that the events aren't (physically) on two different continents anymore.

Oh, good! I was wondering what was happening to QPL 2020. Could you keep me in the loop on QPL, please?

Tobias Fritz (Apr 27 2020 at 15:24):

Aleks Kissinger said:

Hey Tobias Fritz, could you put me on the list as well? I'm now co-organising QPL (which is going online as well), on June 2-6, so would def be interested in try to come to the later part, now that the events aren't (physically) on two different continents anymore.

Great to hear, Aleks! I've put you on the list.

Aleks Kissinger (Apr 28 2020 at 08:32):

Arthur Parzygnat said:

Oh, good! I was wondering what was happening to QPL 2020. Could you keep me in the loop on QPL, please?

Sure. If you are on one of the "usual" mailing lists for QPL announcements (e.g. categories, types, act, quantum-foundations, logic) you'll see an announcement about it. I'll try to post something here as well once details are available.

Arthur Parzygnat (Apr 28 2020 at 09:09):

Aleks Kissinger said:

Arthur Parzygnat said:

Oh, good! I was wondering what was happening to QPL 2020. Could you keep me in the loop on QPL, please?

Sure. If you are on one of the "usual" mailing lists for QPL announcements (e.g. categories, types, act, quantum-foundations, logic) you'll see an announcement about it. I'll try to post something here as well once details are available.

I am not on any such mailing list. Probably because I never attended QPL (I almost went to QPL 2019, but plans fell through, and I was hoping to attend 2020 since I'm currently near Paris). Who should I contact to be placed on such a mailing list?

Evan Patterson (May 27 2020 at 02:35):

For any interested categorical statisticians, I've put up a draft of my PhD thesis: https://www.epatters.org/assets/thesis.pdf

The goal of the previously unpublished material here is to understand statistical models from the viewpoint of categorical logic. I propose a notion of "statistical theory" whose models are statistical models in the conventional sense. This idea is introduced in Sec 1.1 and developed formally in Ch 3, with a large number of examples given in Ch 4. The project builds on work by a number of people in this community.

I've tried to write the thesis in such a way that a statistician could learn something about category theory and a category theorist learn something about statistics. Any feedback is welcome. This is not the final version, so it is not too late for me to make clarifications or corrections if needed.

John Baez (May 27 2020 at 03:23):

Great! I like this thesis.

Evan Patterson (May 27 2020 at 03:38):

Thanks!

Nathanael Arkor (May 27 2020 at 14:40):

I've only skimmed through, but what I've seen looks very interesting. I was wondering whether you've considered what an equational logic for statistical theories could look like? (You may need a slightly stricter variant, analogous to the "strict structure-preserving ioo functor from the free so-and-so category, to get an equivalence between the two notions.)

Nathanael Arkor (May 27 2020 at 14:48):

I should like to understand the notion of "supply of a semilattice of PROPs" better. Are there some examples of "well-known" categories that supply semilattices of PROPs, not motivated by linear algebra?

Nathanael Arkor (May 27 2020 at 14:51):

I also wonder what supplying a coloured PROP might mean.

Nathanael Arkor (May 27 2020 at 15:20):

Is a category homomorphically supplying a PROP $P$ related to the category being equivalent to a category of models for $P$? E.g. something like: "$\mathscr C$ homomorphically supplies a PROP $P$ iff it is equivalent to symmetric monoidal category of models of $P$ with appropriate coherence conditions.".

Evan Patterson (May 27 2020 at 20:36):

Hi Nathanael, thanks for the great questions. If only I could answer all of them :)

I haven't thought about an equational logic for statistical theories. It's a question of practical interest to me, though, since I'd eventually like to implement this stuff. As I mention in the conclusion, it would be natural to hook up an implementation of statistical theories to a probabilistic programming language so that you can sample from and fit models.

Supplies of semilattices of PROPs are intended for situations where you have a family of related theories, such that some theories can be seen as sub-theories of others in a standard way. One example I thought about, but didn't include in the thesis, are the "weak" or "partial" notions of symmetry that you get by relaxing the definition of a group. Because identity and inverse elements are unique whenever they exist, the category of semigroups supplies a semilattice of PROPs for, say, semigroups, inverse semigroups, monoids, inverse monoids, and groups. So that's one additional example. It would be good to come up with more.

Your last question applies to Fong and Spivak's original notion. Seems very plausible. I wonder if Brendan or David have looked into stating it formally. It certainly seems like the right intuition about homomorphic supply.

Jade Master (May 28 2020 at 00:29):

Ooh thanks Evan. I'm excited to get into it.

Evan Patterson (May 28 2020 at 00:43):

Thanks Jade! We talked about an early version of this at Siemens. I fleshed it out quite a bit since then, but there's still much that I don't understand and many possible directions to explore.

Jade Master (May 28 2020 at 01:22):

I'm noticing some categorical discussion of Markov kernels. Did you mention that Markov kernels are morphisms in the Kleisli category of the Giry Monad? I'm pretty sure this is true...but maybe you thought about it and decided it wasn't.

Evan Patterson (May 28 2020 at 01:27):

It's true, and I think I mention it in the Notes to Ch 3, but I don't mention it in the main text. Generally in this work, I try to use only the categorical concepts that I really need, so as to keep the prerequisites as low as possible.

Jade Master (May 28 2020 at 01:28):

Right that makes sense. There's no use in abstract nonsense for it's own sake :)

Jade Master (May 28 2020 at 01:30):

A Markov kernel $m : (X, \Sigma_X) \to (Y,\Sigma_Y)$ being either a map $m : X \times \Sigma_Y \to [0,1]$ or a map $m : X \to \mathrm{Prob} (Y)$ reminds me of how profunctors can be regarded either as functors $X \times Y^{op} \to \mathsf{Set}$ or as functors $X \to \mathsf{Set}^Y$.

Evan Patterson (May 28 2020 at 01:30):

I mean, I don't know if I can get many statisticians to read it, but I can at least try :)

Jade Master (May 28 2020 at 01:31):

When you think of profunctors in the second representation, their composition is given by Kan extension along the Yoneda embedding $X \to \mathsf{Set}^X$

Evan Patterson (May 28 2020 at 01:32):

Oh interesting, I didn't think about a possible connection to profunctors.

Jade Master (May 28 2020 at 01:32):

I'm curious about whether or not the formula for composition of Markov Kernels can be thought about in a similar way.

Jade Master (May 28 2020 at 01:35):

Evan Patterson said:

Oh interesting, I didn't think about a possible connection to profunctors.

Profunctors can also be thought of in terms of Kleisli categories...except that everything is 2-dimensional. Prof is the Kleisli bicategory of the completion by presheaves (2-monad?) (this is probably more categorical nonsense then is necessary at all but it's nice to know about maybe)

Jade Master (May 28 2020 at 01:35):

@Christian Williams taught me about this. I bet I made a few technical mistakes in the statement.

Christian Williams (May 28 2020 at 01:42):

yeah! looks right to me. (Nathanael and I like Kleisli bicategories as a framework for generalized operads and type theory; but that's off-topic here.)

Jade Master (May 28 2020 at 02:09):

For Lawvere theories, a very powerful fact that you get is that for a morphism of Lawvere theories $f : Q \to R$, the pullback functor $\mathsf{Mod}(R) \to \mathsf{Mod}(Q)$ has a left adjoint. Do you think that something similar may be true for models of statistical theories? For Lawvere theories, you can use this fact to prove facts about completeness, and to get constructions of free models. Maybe you could get similar results for statistical theories...

Evan Patterson (May 28 2020 at 02:20):

Great point. I wish I had something in the thesis about that but I don't. I'd love to know the answer. An obstacle is that Markov doesn't have anywhere near the limits or colimits that Set does. It does at least have coproducts. (IIRC that is a general property of Kleisli categories.)

Oliver Shetler (May 28 2020 at 17:12):

I'm very excited to read this paper @Evan Patterson.

Evan Patterson (May 28 2020 at 17:22):

Thanks!

Tobias Fritz (May 29 2020 at 21:14):

The #Categorical Probability and Statistics 2020 workshop is less than a week away! We have now posted a schedule which should be close to final.

I will be sending out an email to participants this evening with the Zoom link to where the action will be happening. If you'd like to participate but are not on the email list (and you're not a speaker), then just PM or email me, @Rory Lucyshyn-Wright or @Paolo Perrone and we will be in touch. The live stream on YouTube will be viewable either way, and we will be posting the link to it by early next week.

Oliver Shetler (Jun 14 2020 at 13:51):

Here is the link for the Categorical Statistics reading group.

You can also find other meetings at https://www.meetup.com/Category_Theory/

Notification Bot (Nov 17 2021 at 18:49):

This topic was moved by Jules Hedges to #general: off-topic > probability and statistics