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Stream: theory: categorical probability

Topic: Categorical imprecise probability

Sam Staton (Nov 12 2024 at 11:05):

Hi, Jack and I have a paper accepted to the POPL conference about imprecise probability in a categorical setting. Imprecise probability is, roughly, uncertainty about which probability distributions to use. It's sometimes modelled by sets of distributions. (It was also one topic of the adjoint school this summer.)

Our main categorical idea is this: there are known impossibility theorems for an algebraic approach, including Eckmann-Hilton-like arguments, but we can sidestep them by using graded theories / graded Markov categories (i.e. presheaf enriched), and converting some of the equations into regradings (morphisms). Happy to discuss. If you have any comments on the draft, we're grateful to hear them, and we can still make edits to the paper this week.

https://arxiv.org/abs/2405.09391

Benedikt Peterseim (Nov 19 2024 at 20:15):

I found it interesting that you show the “convex subsets of distributions” monad is not commutative, despite earlier claims that this is so in the literature. Is this somehow inevitable, or could there potentially also be a version of this that violates Desideratum 2, but not Desideratum 1?

Benedikt Peterseim (Nov 19 2024 at 20:16):

(Sorry if I didn’t read carefully enough and this is already answered in the paper.)

Benedikt Peterseim (Nov 19 2024 at 20:21):

By “a version of this”, I mean some commutative affine monad of “certain kinds of subsets of certain kinds of probability measures” (without keeping track of naming). So in other words, my question is if there any hope left that such monad exists?

Sam Staton (Nov 19 2024 at 23:39):

Thanks Benedikt. There is a model that violates Desideratum 2 (good coproducts, basically), which is usual "connectedness" quotient involved in the "Para" construction. However, this would not be a monad on a distributive category. I wonder whether this could still be useful in some situations.

Sam Staton (Nov 19 2024 at 23:41):

On Set, there can be no commutative affine monad that contains the finite powerset and probability monads. This is an Eckmann-Hilton-like argument that I wrote up on nlab a while ago here . (I would be interested to know whether there is a precise sense that this is like Eckmann-Hilton!)

Sam Staton (Nov 19 2024 at 23:43):

Let me know if I misunderstood your question. There are some subtle points around.

Benedikt Peterseim (Nov 20 2024 at 05:43):

Thanks for your reply! This does answer my question, especially your second paragraph. (I will try to understand this Eckmann-Hilton-like argument better.)

Benedikt Peterseim (Nov 20 2024 at 05:45):

Very nice overview section, pictures and examples, by the way! Makes it conceptually very clear.

Ralph Sarkis (Nov 26 2024 at 15:22):

Could you clarify what you mean by the "monoidal product of morphisms is also natural"? I would summarize it as follows:
image.png

This does not imply the interchange identity because $f$, $f'$, $g$, and $g'$ are all restricted to the grade $I$ (the monoidal unit). It makes sense because in general the interchange identity does not typecheck: for instance, $(1 \otimes h_1) ; (h_2 \otimes 1)$ has grade $a \otimes b$ while $(h_1 \otimes h_2)$ has grade $b \otimes a$. However, I would still like to prove $(h_1 \otimes 1) ; (1 \otimes h_2) = h_1 \otimes h_2$, but I don't see how to do that.

Sam Staton (Nov 26 2024 at 17:24):

Hi Ralph, Thanks for this and for your other input. Really grateful. There is a version later than the arxiv one and we are still fixing up the final version, including your suggestions, and hopefully this will be clearer.

Sam Staton (Nov 26 2024 at 17:25):

For now: Since you're probably happy with enriched categories, a "${\mathbb{G}}$-graded monoidal category" is just a $\hat{\mathbb{G}}$-enriched monoidal category with $\hat{\mathbb{G}}$ regarded with the Day tensor. NB I mean monoidal as an enriched category. This does have interchange modulo regrading, by definition.
I think you're looking at Definition 2.3, which unpacks the data for that without mentioning presheaves. (Those come in Proposition 3.5, which you could take as the definition if you're happy with enriched categories.) Apologies, the conditions in Def 2.3 in the arxiv version were too fast, I agree.