Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: event: MIT Categories Seminar

Topic: July 23: Evan Patterson's talk

Paolo Perrone (Jul 21 2020 at 12:49):

Hi all! This is the official discussion thread for Evan's talk, "The algebra of statistical theories and models".
When: Thursday July 23rd, 12:00 noon EDT (Boston time)
Zoom meeting: https://mit.zoom.us/j/280120646
YouTube live stream: https://youtu.be/IqIpELL7Mj0

Paolo Perrone (Jul 23 2020 at 15:45):

Hello all! We start in 15 minutes.

Paolo Perrone (Jul 23 2020 at 16:53):

Main reference (as given by Evan): https://arxiv.org/abs/2006.08945

Paolo Perrone (Jul 23 2020 at 17:13):

We are virtually hanging out here! https://gather.town/8u0TXRzLpBsKwHjz/mit-categories-seminar

Evan Patterson (Jul 23 2020 at 17:14):

The room is password protected. Can you share the password?

Evan Patterson (Jul 23 2020 at 17:15):

Nevermind, it is: "yonedalemma"

Tobias Fritz (Jul 23 2020 at 17:37):

Very nice talk!

Here's a question aimed at understanding models of statistical theories in general (and relating them to what I said last week about the possibility of developing categorical algebra for hyperstructures). What is a linear model in $\mathbf{Vect}$, considered as a monoidal category with respect to cartesian product (=direct sum) and with the obvious supplies? Or does this question not make sense because something goes wrong with the supplies?

Tobias Fritz (Jul 23 2020 at 17:48):

Asking slightly differently, does it make sense to try and "collapse" every statistical theory to a multisorted algebraic theory by forcing every morphism to be deterministic? By just imposing determinism of generating morphisms as an additional set of relations? Then what do you get upon collapsing the linear model like this?

Evan Patterson (Jul 23 2020 at 18:48):

Thanks Tobias, this question is interesting. So let's consider a cartesian category $\mathsf{S}$ with the same objects and supply as the Markov category $\mathsf{Stat}$ but with functions/deterministic kernels as morphisms. Then it makes sense to talk about a model of a statistical theory $\mathsf{T}$ in $\mathsf{S}$ as a supply preserving functor $M: \mathsf{T} \to \mathsf{S}$. When $\mathsf{T}$ is the theory of a linear model from the talk, then the "random component" $M(q)$ is forced to be a normal family with zero variance, hence it will have the form $M(q)(x, \sigma^2) = \mathcal{N}(Ax, \sigma^2 \cdot 0) = Ax$. So the linear model collapses into a linear function, with the dispersion parameter $\sigma^2$ playing no role.

Tomáš Gonda (Jul 23 2020 at 20:09):

Thanks for the talk @Evan Patterson, I feel like I am starting to understand this somewhat.

One basic thing that I don't understand is the construction where you go from a lattice of SMC interpolating between sets and vector spaces to the lattice of PROPs. I guess I am not really sure what the vertices of the latter, like $\mathsf{Th(Conv)}$, are, not just just how they are generated.

Tobias Fritz (Jul 23 2020 at 20:29):

Right, that makes sense @Evan Patterson. Thanks!

Paolo Perrone (Jul 23 2020 at 20:30):

Video here!
https://youtu.be/Kzl2N9SH6H8

Evan Patterson (Jul 23 2020 at 20:44):

Tomáš Gonda said:

One basic thing that I don't understand is the construction where you go from a lattice of SMC interpolating between sets and vector spaces to the lattice of PROPs. I guess I am not really sure what the vertices of the latter, like $\mathsf{Th(Conv)}$, are, not just just how they are generated.

No problem, I didn't explain that in any detail. The vertices in the latter are PROPs (i.e., strict SMCs where the monoid of objects is freely generated by one object). We think of them as theories. So, for example, $\mathsf{Th(Conv)}$ has a commutative comonoid $x \to x \otimes x$ and $x \to I$ for copying and deleting, plus a morphism $x^{\otimes n} \to x$ for every convex combination $(a_1,\dots,a_n) \in \Delta^{n-1}$, which obey some axioms. A model of this theory, namely a symmetric monoidal functor $\mathsf{Th(Conv)} \to \mathsf{Set}$, is a convex space. This is an object of the category $\mathsf{Conv}$, hence the notation.

A supply of this lattice of PROPs in an SMC $\mathsf{C}$ is then an assignment of a PROP, plus a model of that PROP, at every object in $\mathsf{C}$ in a way that is compatible with the monoidal product of $\mathsf{C}$.

Tomáš Gonda (Jul 23 2020 at 21:07):

ok thanks, I think that makes sense. Somehow, I thought from the way you presented it that there should be a concrete construction that takes a lattice of SMC and generate the lattice of PROPs, but it seems a bit more nuanced. Staying with convex spaces, is there a way to generate $\mathsf{Th(Conv)}$ from $\mathsf{Conv}$?

Evan Patterson (Jul 23 2020 at 21:32):

Right, in that sense the notation is misleading because I do not treat $\mathsf{Th}$ literally as a function from categories to theories. In my thesis, I write down the theories explicitly. But, rather amazingly, I think it is possible to take this notation literally, because for any algebraic theory, the theory can be reconstructed from its category of models. This is sometimes called "Lawvere duality." Other people around here could explain this much better than me, or you can read about it in these lecture notes by Awodey and Bauer. See also this paper.