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Tobias Fritz (Jun 06 2020 at 18:00):I can comment on Kolmogorov's extension theorem in the context of Markov categories, although my interpretation of the question was also that Peter was referring to something else. There are many versions of the extension theorem which differ in the precise assumptions, but the general idea is that joint distributions of an infinite family of random variables are in bijection with compatible families of finite marginals. In other words, the probability measures on an infinite product space correspond exactly to families of probability measures on all finite subproduct, together with the compatibility condition that marginalizing further to a subsubproduct produces exactly the given measure there. (Actually I now realize that this is vaguely like the inheritance that Peter had talked about yesterday.) Categorically, this turns into the idea that one can define infinite products in a Markov category as the filtered limits of their finite monoidal subproducts. This is the essential idea behind the definition of infinite tensor product in my paper with Eigil (Definition 3.1). In other words, we turn (a version of) the Kolmogorov extension theorem into a definition.
There are some important subtleties here having to do with the topologies and regularity conditions coming up in the general form of the theorem. My hope is that there is a Markov category in which those hold by definition. Perhaps one has work with enriched Markov categories in order to make this work.
Alex Simpson (Jun 06 2020 at 18:46):Building on Tobias' comment, in the case of the category of Markov kernels on measurable spaces, Kolmogorov extension can be formulated as: the "Kleisli inclusion" functor from Set to Markov kernels preserves certain cofiltered limits. I wonder if it even preserves all cofiltered limits!? (If not, it would be good to have an understanding of the class of projective limits that is preserved.)
Sam Staton (Jun 06 2020 at 20:27):Alex Simpson said:
The "Kleisli inclusion" functor from Set [Meas?] to Markov kernels preserves certain cofiltered limits. I wonder if it even preserves all cofiltered limits!? (If not, it would be good to have an understanding of the class of projective limits that is preserved.)
I agree. In Dirichlet is natural, Theorem 2.5, Vincent and Ilias say that Giry preserves limits of countable surjective cofiltered diagrams on Polish spaces, and they call this "Bochner extension". So we know that much at least.
Tobias Fritz (Jun 06 2020 at 21:09):@Alex Simpson, @Sam Staton: that touches upon one of my favourite puzzles in categorical probability! Namely, is there a probability monad on a class of spaces which everyone can come to regard as the probability monad, eliminating the need to distinguish between various additional niceness properties of measures like -smoothness or tightness / inner regularity? Then I would (perhaps naively) expect the underlying functor of this monad to preserve all cofiltered limits.
With the probability measures functor on measurable spaces, one can perhaps not expect more than the preservation of countable cofiltered limits. But probability functors on other categories of spaces can preserve more general cofiltered limits, which in my perspective is an important argument for working with such spaces instead of just . This is closely related to tightness (or inner regularity) of the measures involved. For example, consider the Radon monad, whose underlying functor assigns to every compact Hausdorff space the space of Radon probability measures on it. Then it's not hard to prove, using the Riesz representation theorem and the Stone-Weierstraß theorem, that this preserves all cofiltered limits. But the Radon monad is about as nice as it gets! And is too limited a category of spaces for the Radon monad to be the ultimate probability monad.
For continuous valuations on arbitrary spaces, there are some very nice results on cofiltered limits by Goubault-Larrecq, which give necessary and sufficient conditions for the Kolmogorov extension to exist. This seems closely related to the duality of open and compact sets in topology, and my suspicion is that tightness and regularity conditions on measures will one day find their most natural home in a probability monad on a category of spaces built around this duality.
Tobias Fritz (Jun 06 2020 at 21:18):Note: I've made this into its own topic, since it seems separate from Evan's talk.
Alex Simpson (Jun 07 2020 at 06:14):@Tobias Fritz You say "With the probability measures functor on measurable spaces, one can perhaps not expect more than the preservation of countable cofiltered limits". But the Kolmogorov extension theorem itself, in which one constructs a measure on for , involves uncountable diagrams, and their limits are preserved!
Sam Staton (Jun 07 2020 at 08:22):Thanks Tobias Fritz, that's helpful.
@Alex Simpson I remembered Rao's paper projective limits of probability spaces, which gives necessary and sufficient conditions (Section 4), wonder if that is along the lines you had in mind.
Alex Simpson (Jun 07 2020 at 14:51):@Sam Staton Thank you for that reference. That indeed looks like a pretty thorough analysis of the situation.
Juan Pablo Vigneaux (Jun 07 2020 at 21:29):Tobias Fritz said:
I can comment on Kolmogorov's extension theorem in the context of Markov categories, although my interpretation of the question was also that Peter was referring to something else. There are many versions of the extension theorem which differ in the precise assumptions, but the general idea is that joint distributions of an infinite family of random variables are in bijection with compatible families of finite marginals. In other words, the probability measures on an infinite product space correspond exactly to families of probability measures on all finite subproduct, together with the compatibility condition that marginalizing further to a subsubproduct produces exactly the given measure there. (Actually I now realize that this is vaguely like the inheritance that Peter had talked about yesterday.) Categorically, this turns into the idea that one can define infinite products in a Markov category as the filtered limits of their finite monoidal subproducts. This is the essential idea behind the definition of infinite tensor product in my paper with Eigil (Definition 3.1). In other words, we turn (a version of) the Kolmogorov extension theorem into a definition.
There are some important subtleties here having to do with the topologies and regularity conditions coming up in the general form of the theorem. My hope is that there is a Markov category in which those hold by definition. Perhaps one has work with enriched Markov categories in order to make this work.
@Tobias Fritz I may be misunderstanding something, but I believe that the problem here is not at the level of definitions. The projective systems of probability spaces were already considered in the "classical" references (as Rao nicely explains) but the whole problem was the existence of the limit in the category under consideration (or the "representability" of the limit, if one adopts the terminology of the SGA4). I doubt that existence could be proved in a purely synthetic way.
Tobias Fritz (Jun 07 2020 at 21:58):@Juan Pablo Vigneaux: I think that everyone agrees about the definition of probability measure on well-behaved spaces like Polish spaces. But how do we know what the "correct" definition of probability measure is on less well-behaved spaces, such as uncountable product spaces, spaces with bad separation, etc? Then the definition of probability measure branches into a number of non-equivalent notions. Already the standard definition of probability measure can be applied to the Borel -algebra or to the Baire -algebra, which are different in general. Then there are additional properties like -smoothness or inner regularity which a given probability measure may or may not have. So I think that yes, it is a problem at the level of definitions to some extent. And the preservation of all cofiltered limits may (or may not) be one criterion for a good definition. As I've already mentioned, the Radon monad achieves this (but has other deficiencies).
A related issue is that the Borel -algebra functor from (say) Hausdorff spaces to measurable spaces does not preserve uncountable products. For example, if we naively took an uncountable product of with itself in order to describe the joint distribution of uncountably many real-valued random variables, then we would not be able to talk about the probability of all of those variables taking the same value, since the diagonal would not be a measurable set! This seems clearly undesirable to me. (In practice the diagonal will still be Caratheodory measurable, but that only illustrates my point: the product -algebra is too small to be the "correct" one.)
In summary, I don't think that we should consider the definition of probability space to be set in stone. In particular, measure theory often has aspects which involve not just a measurable structure, but also a topology on the underlying space, for example for uncountable products or for talking about supports. And I think that when constructing probability monads beyond Polish spaces or standard Borel spaces, we should take that topological structure into account. Then the question arises of what the optimal way to do that will be.
Does this seem reasonable to you? Apologies if my thoughts on this aren't so well-organized yet.
Juan Pablo Vigneaux (Jun 07 2020 at 22:27):There is definitely a misunderstanding then haha I think it's because I was strictly referring to your initial comment, in relation to the title of this thread, and not to the other ideas that you have discussed here. What I was trying to say is: although one can define "infinite products in a [a given] Markov category as the filtered limits of their finite monoidal subproducts", the main problem that Kolmogorov's theorem addresses is the existence of such product inside the initial category under consideration (since one could also introduce clever "minimal" enlargements of that category). Hence for any particular Markov category, one will need a Kolmogorov theorem, unless there is a synthetic proof of such theorem. Of course, I understand that one can make an "axiom" that says "filtered limits exist i.e. Kolmogorov's theorem holds", but this is very different from actually verifying that it holds in a given category, which is what Kolmogorov did.
Concerning your reply, I think you're right: there are several notions of "well-behaved" measures and the "good" definition is rather elusive, and I also think that category theory could be a good tool to understand better what's going on.
Tobias Fritz (Jun 07 2020 at 22:33):Oh, I see! Yes, our synthetic treatment of infinite products turns the theorem into a definition, and one still needs to prove that the theorem holds in the category under consideration (or realize that it doesn't).
You're right that the title of this thread is a bit misleading too, also in light of the fact that some of those extension theorems are associated with other names like Bochner or Prokhorov. I'll fix that.