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Stream: event: Categorical Probability and Statistics 2020 workshop

Topic: Perspectives and open problems

Tobias Fritz (Jun 08 2020 at 22:24):

It has been suggested that we should have a topic to list perspectives and open problems on categorical probability and statistics. Here we go! Please feel free to add your thoughts.

Paolo Perrone (Jun 08 2020 at 22:43):

Very good! I start: do random variables form a monad, in any category and any context?

Tomáš Gonda (Jun 09 2020 at 02:16):

Related to the talks we saw this week - there were at least three accounts of conditional expectations in the talks of Paolo Perrone, Alex Simpson, and Prakash Panangaden. Is there a way to view these from a common perspective, or alternatively, is there a way to translate between these points of view; at least in some specific context if not in general? Connected to this is the question of whether Markov categories lend themselves to an account of conditional expectation, broadly construed.

Arthur Parzygnat (Jun 09 2020 at 09:33):

One of the open questions I raised during my talk was the following:
"Is every S-positive subcategory of finite-dimensional $C^*$ -algebras and unital maps contained in the category of finite-dimensional $C^*$ -algebras and Schwarz positive unital maps?''
This is interesting from both the physical and operator-algebra perspective since if true, it would provide an alternative (diagrammatic) definition of Schwarz positivity. Taking the subclass of morphisms closed under the monoidal product would then give an alternative definition for CPU maps. If false, how can the counter-example be interpreted? What is its physical content? Is positivity appropriate terminology for such a category?

Paolo Perrone (Jun 09 2020 at 13:26):

Tomáš Gonda said:

Related to the talks we saw this week - there were at least three accounts of conditional expectations in the talks of Paolo Perrone, Alex Simpson, and Prakash Panangaden. Is there a way to view these from a common perspective, or alternatively, is there a way to translate between these points of view; at least in some specific context if not in general? Connected to this is the question of whether Markov categories lend themselves to an account of conditional expectation, broadly construed.

@Tomáš Gonda Very good point. I'm quite confident that Prakash' construction and mine are very compatible, but I still have to figure out the details. (This may be easier if we can find a monad of random variables). I haven't understood Alex' construction yet, but it's on my to-do list to rewatch the video very carefully.
I believe that Markov categories with conditionals are a good starting point to talk about conditional expectation in a unified way.

Tomáš Gonda (Jun 09 2020 at 16:46):

Here's another (admittedly quite vague) question that I have been thinking a bit about following the workshop:

Given a category $\mathsf{Caus}$ that models a situation of interest (deterministically?) and a suitable Markov category $\mathsf{Inf}$ that describes one's theory of uncertainty and/or inference, is there a generic way to construct a causal-inferential theory $\mathsf{C-I}$ as presented by @Rob Spekkens? Does imposing inferential equivalence yield another Markov category?
Conversely, given a Markov category $\mathsf{C}$ with suitable properties (cartesian closed?; possessing strong enough version of the randomness pushback axiom), can one construct a causal-inferential theory in which the $\mathsf{Caus}$ corresponds to the deterministic subcategory of $\mathsf{C}$ ?

Sam Staton (Jun 09 2020 at 18:10):

A question I've been bothered by for three or four years: is the category of s-finite kernels a Kleisli category for a monad on measurable spaces? (An s-finite kernel is a countable sum of subprobability kernels; these are all the kernels formed by composing probability kernels and one-point measures, as in my tutorial.)

John Baez (Jun 09 2020 at 19:45):

That's an interesting puzzle!

Arthur Parzygnat (Jun 10 2020 at 07:57):

Robert Furber said:

The problem is that we are not guaranteed to get a normal homomorphism from a measurable transformation, only a sigma-normal one. This is really the most difficult part of the deleted chapter of my thesis. The way of fixing it in the case of compact strictly localizable spaces is given here:
https://mathoverflow.net/a/361787/61785

@Robert Furber, I'm only beginning to explore this area and I am curious about this paragraph. What is the precise statement? If $(X,\Sigma,\mu)\xrightarrow{f}(Y,\Lambda,\nu)$ is a measure-preserving function between measure spaces, then the corresponding homomorphism $L^{\infty}(Y,\Lambda,\nu)\xrightarrow{L^{\infty}(f)}L^{\infty}(X,\Sigma,\mu)$ (obtained via pullback) need not be normal? If so, under what assumptions on the measure spaces guarantee normality? (Simply pointing me to a reference would be incredibly helpful for me).
@Tobias Fritz, yes to your question from earlier, I was hoping to start exploring the ideas for $W^*$ -algebras and/or von Neumann algebras, but now I'm a little worried this may be overwhelmingly technical... Ideally, I would like a reasonably concise (not overly technical, but correct) reference discussing Markov kernels between probability spaces and the corresponding discussion for abelian von Neumann algebras. Since I'll be working with probability measures, which are finite (and hence $\sigma$ -finite), I was under the impression that it doesn't matter what the underlying measurable structure is for most (all?) results that I would need.

Alex Simpson (Jun 10 2020 at 09:17):

Tomáš Gonda said:

Related to the talks we saw this week - there were at least three accounts of conditional expectations in the talks of Paolo Perrone, Alex Simpson, and Prakash Panangaden. Is there a way to view these from a common perspective, or alternatively, is there a way to translate between these points of view; at least in some specific context if not in general? Connected to this is the question of whether Markov categories lend themselves to an account of conditional expectation, broadly construed.

As I recall, Paolo explained the classic definition of conditional expectation $E(X \mid \mathcal{F})$ , due to Kolmogorov. Prakash also started his talk with the same definition, and made the connection himself to his (and coauthors') functorial version.

The definition in my talk is formulated in my peculiar axiomatic world, so is more difficult to directly compare to the standard definition. Nevertheless the ideas behind it can be described in the traditional Kolmogorov setting, where it corresponds to a special case of conditional expectation a là Kolmogorov. Let $(A, \Sigma_A)$ be a measurable space, $(\Omega, \Sigma_\Omega, P_\Omega)$ a probability space and let $Y : \Omega \to A$ be a measurable function. Suppose also that $\Sigma_A$ contains all singletons and $\mathsf{image}(Y)$ . Then the following are equivalent for a function $Z : \Omega \to \mathbb{R}$ .

$Z$ is $Y^{-1}\Sigma_A$ measurable (where $Y^{-1}\Sigma_A$ is the sub- $\sigma$ -algebra of $\Sigma_\Omega$ consisting of all sets $Y^{-1}B$ for $B \in \Sigma_A$ )
There exists a measurable $f: A \to \mathbb{R}$ such that $Z = f \circ Y$ .

This means, that under the situation for $Y$ described above, the usual Komolgomorov definition of $E(X \mid Y)$ , can be reformulated by replacing the condition " $E(X \mid Y)$ is $Y^{-1}\Sigma_A$ -measurable" with "there exists measurable $f: A \to \mathbb{R}$ such that $E(X \mid Y) = f \circ Y$ " (and the equality $E(X \mid Y) = f \circ Y$ can also equivalently be replaced with almost-sure equality) - that is, "functional dependency" is equivalent to the usual measurability condition.

In the traditional setting, this special case does not enjoy the generality of the full Kolmogorov definition. In my setting, this special case seems to be powerful enough to do everything one wants to do.

Paolo Perrone (Jun 10 2020 at 14:34):

This way of phrasing conditional expectation, using a function rather than a sub-sigma-algebra, is also the way I like seeing it. Moreover, it highlights the connection with Kan extension even more.