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Stream: theory: categorical probability

Topic: unnormalised Markov categories with conditionals

Nathaniel Virgo (Jun 06 2022 at 03:00):

I've recently found some nice uses for unormalised Markov categories, aka Copy-Discard categories. That is, categories that obey all the axioms of a Markov category except that we don't require all morphisms to commute with the delete map.

I would like to know if there is a nice known example of such a category where (1) a morphism $A\to B$ has the form of a map from $A$ to measures that need not be probability measures over $B$, and (2) it has conditionals, in the sense of Fritz' definition 11.5 (but with "Markov category" replaced with "unnormalised Markov category")

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The category of unnormalised Markov kernels between finite sets has this property, and I'm hoping to find a nice analog for the measure-theoretic case.

If such a thing is known I might have some nice applications for it, but I'm not really a measure theory person, so I'm hoping that not only is an example known but also a bunch of its category-theoretic properties are known, so I can work with it as a category rather than needing to do proofs at the measure-theoretic level.

Tobias Fritz (Jun 06 2022 at 06:53):

Hi! The one category that I know of in the literature is @Sam Staton's category of s-finite kernels, introduced in his paper Commutative semantics for probabilistic programming. See also On S-Finite Measures and Kernels for a review of the relevant measure theory. Sam describes it as a monoidal category, but I think that it becomes a copy-discard category in the obvious way.

Actually technically one gets a simpler category if one restricts to finite kernels. Then a morphism between measurable spaces $X \to Y$ is a map $f(\cdot|\cdot) : \Sigma_Y \times X \to [0,\infty)$ such that $f(-|x)$ is a finite measure on $Y$ for every $x$ whose total mass $f(Y|x)$ is bounded in $x$, and such that $f(A|\cdot) : X \to [0,\infty)$ is measurable for every $A \in \Sigma_Y$. These kernels compose as usual via Chapman-Kolmogorov and should form a copy-discard category in the obvious way.

Concerning conditionals, it seems unlikely to me that either of these categories would have conditionals, since there are too many pathological measurable spaces; already with normalization one doesn't have conditionals (the case of $\mathsf{Stoch}$). So we usually restrict to nice measurable spaces, namely the standard Borel ones. Up to isomorphism, these are only the countable discrete ones and $\mathbb{R}$. The resulting Markov category $\mathsf{BorelStoch}$ has conditionals, but the proof already seems to be quite involved. I'm not sure if the restriction to standard Borel spaces would result in conditionals for either or both of the unnormalized categories I described.

I bet that Sam can tell us more!

Tobias Fritz (Jun 06 2022 at 07:04):

If I'm interpreting it correctly, Kallenberg's Stationary and invariant densities and disintegration kernels referenced by Sam shows that the category of s-finite kernels between standard Borel spaces does at least have conditional distributions (Lemma 2.2 in there). So that looks promising for s-finite kernels.

Sam Staton (Jun 06 2022 at 08:45):

Thanks, yes someone should figure out how this disintegration in CD-categories fits into categorical frameworks. There is also a discussion of disintegration of s-finite kernels in Section 9 of this article by @Matthijs Vákár and @Luke Ong. It is curious that they take care over $\infty$-sets, which don't seem to be mentioned in the Kallenberg lemma that you cited. Maybe that would factor in to the categorical analysis nicely.

Sam Staton (Jun 06 2022 at 08:49):

Ah, we also discussed it here. How time flies.

Tobias Fritz (Jun 06 2022 at 08:58):

Oh, right :astonished: I had already forgotten about that...!