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: ACT@UCR

Topic: April 8th: Prakash Panangaden

Joe Moeller (Apr 02 2020 at 16:28):

Next week's talk will be given by Prakash Panangaden

Title: A categorical view of conditional expectation

Abstract: This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones---which are abstract versions of the familiar cones in vector spaces---of measures and related categories cones of Lₚ functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

John Baez (Apr 02 2020 at 17:23):

This talk is based on the following paper:

• Philippe Chaput, Vincent Danos, Prakash Panangaden and Gordon Plotkin, Approximating Markov processes by averaging, in _International Colloquium on Automata, Languages, and Programming_, Springer, Berlin, 2009.

John Baez (Apr 03 2020 at 19:30):

Prakash says those two versions are the same.

John Baez (Apr 06 2020 at 20:21):

You can see Prakash's slides now:

• Prakash Panagaden, A categorical view of conditional expectation.

Abstract. This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones—which are abstract versions of the familiar cones in vector spaces—of measures and related categories cones of Lp functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

Don't forget to join us Wednesday April 8th at 5 pm UTC, which is 10 am in California, 1 pm on the east coast of the United States, or 6 pm in England. The talk will be held online via Zoom, here:

https://ucr.zoom.us/j/607160601

We will have discussions online here—I suggest coming here 20 minutes before the talk, for coffee and delicious cookies.

People will also discuss the talk afterwards.

Joe Moeller (Apr 08 2020 at 15:29):

Prakash's talk will start today at UTC 1700 = 10am in CA = 6pm in the UK.

sarahzrf (Apr 08 2020 at 15:30):

!

John Baez (Apr 08 2020 at 15:33):

Prakash's talk will be held online via Zoom, here:

https://ucr.zoom.us/j/607160601

Paolo Perrone (Apr 08 2020 at 15:33):

This sounds so interesting! Can't wait to hear.

John Baez (Apr 08 2020 at 15:34):

The slides, again, are here:

• Prakash Panangaden: A categorical view of conditional expectation.

Eigil Rischel (Apr 08 2020 at 16:00):

I'm being asked for a meeting password?

Joe Moeller (Apr 08 2020 at 16:03):

@John Baez I was also asked for a password.

John Baez (Apr 08 2020 at 16:29):

Okay, yeah that was imposed by UCR. Wait a sec.

John Baez (Apr 08 2020 at 16:31):

I've turned off the need for a password, but it's 632765 in case you need it.

Ian Price (Apr 08 2020 at 16:34):

Given the increase in zoombombing, it's a sensible precaution

John Baez (Apr 08 2020 at 16:35):

Right. I think I'll risk it now.

Min Ro (Apr 08 2020 at 16:35):

Given that this is a talk, is there a way so that only a select few people can use cam/audio?

John Baez (Apr 08 2020 at 16:36):

It's set up so that only I can share my screen right now.... and I'll give this privilege to Prakash.

Blake C. Stacey (Apr 08 2020 at 16:40):

The link for the Zoom meeting on the n-Cafe post seems broken; it should point to https://ucr.zoom.us/j/607160601 but has extra Markdown stuff

John Baez (Apr 08 2020 at 16:43):

Whoops, I'll fix it.

John Baez (Apr 08 2020 at 16:44):

Prakash has updated his slides and the new version is here:

http://math.ucr.edu/home/baez/mathematical/ACTUCR/Panagaden_Conditional_Expectation.pdf

John Baez (Apr 08 2020 at 17:03):

It's started...

sarahzrf (Apr 08 2020 at 18:06):

i'm chewing some more on the definitions of cones—is it just me or do ω-complete ones get a unique action of R+ automatically if you have an action of Q

sarahzrf (Apr 08 2020 at 18:06):

actually wait uhh... hmm

sarahzrf (Apr 08 2020 at 18:07):

(basically i'm wondering how much of this is property as opposed to structure)

Oscar Cunningham (Apr 08 2020 at 18:08):

Is $\omega$-continuity stronger or weaker than continuity?

sarahzrf (Apr 08 2020 at 18:08):

if by continuity you mean scott-continuity, it's weaker—dunno if it's strictly weaker

Oscar Cunningham (Apr 08 2020 at 18:09):

I was thinking that the norm on a cone would induce a topology

Oscar Cunningham (Apr 08 2020 at 18:09):

I don't know if that's the same

sarahzrf (Apr 08 2020 at 18:10):

there's a thing on a dcpo (directed-complete partial order) called the scott topology, for which iirc the continuous functions are the same as the scott-continuous ones wrt the order

sarahzrf (Apr 08 2020 at 18:10):

that's what i was thinking of

sarahzrf (Apr 08 2020 at 18:11):

but i think the question of whether that kind of continuity automatically coincides with what you were saying is probably very much in line with what i was asking in the first place

Prakash Panangaden (Apr 08 2020 at 18:11):

Yes, the topology induced by the order is the Scott topology.

Tobias Fritz (Apr 08 2020 at 18:12):

Concerning Arthur's question, I think it's interesting to think about it already when both measurable spaces are finite. Then one difference is that in Prakash's slides, alpha is just a map between the underlying sets, so there aren't really general Markov kernels around. Right?

But I was also wondering about whether there's any relation to the dagger functor of https://arxiv.org/abs/1803.02651

Prakash Panangaden (Apr 08 2020 at 18:12):

No, it is a map between the measurable spaces and it has to do the right things to the measures.

John Baez (Apr 08 2020 at 18:13):

Someone asked if $\omega$-continuity was stronger or weaker than (norm) continuity.

John Baez (Apr 08 2020 at 18:13):

Oscar Cunningham.

Prakash Panangaden (Apr 08 2020 at 18:13):

Stronger in this case.

Tobias Fritz (Apr 08 2020 at 18:13):

Prakash Panangaden said:

No, it is a map between the measurable spaces and it has to do the right things to the measures.

yes, of course, I was referring to the case of finite and discrete measurable spaces for simplicity. So my point was just that it's a very special type of Markov kernel, namely a "deterministic" one

Prakash Panangaden (Apr 08 2020 at 18:15):

Yes, but see the paper to see how we use it to construct approximate Markov kernels.

John Baez (Apr 08 2020 at 18:15):

@Oscar Cunningham - yes, the norm on the cone induces a topology, and so Prakash is saying $\omega$-continuity of maps between cones is stronger (in the current context) than continuity with respect to the norm topology.

Oscar Cunningham (Apr 08 2020 at 18:16):

Thanks!

John Baez (Apr 08 2020 at 18:16):

However I have a feeling that examples that illustrate the difference can only be constructed using the axiom of choice!

Prakash Panangaden (Apr 08 2020 at 18:16):

Hey, Tobias, are you holding the PI workshop on categorical probability via Zoom?

Prakash Panangaden (Apr 08 2020 at 18:18):

Very likely. Examples showing that L_1 is not L_inf^* can only be constructed with the axiom of choice.

Tobias Fritz (Apr 08 2020 at 18:18):

Prakash Panangaden said:

Hey, Tobias, are you holding the PI workshop on categorical probability via Zoom?

Yes, we're planning on a combination of Zoom and Youtube streaming (optional per speaker's choice) as for ACT; Paolo Perrone is helping us with that. But I'm also curious to see how this chat format works out.

Prakash Panangaden (Apr 08 2020 at 18:19):

I think I am not associating my responses correctly with the questions!

John Baez (Apr 08 2020 at 18:19):

The only way I know how to do that is quote the question; you can click on a comment and quote it.

John Baez (Apr 08 2020 at 18:20):

There's not enough "threading" in this discussion group for my taste.

Prakash Panangaden (Apr 08 2020 at 18:20):

John Baez said:

The only way I know how to do that is quote the question; you can click on a comment and quote it.

Like this?

John Baez (Apr 08 2020 at 18:20):

Prakash Panangaden said:

John Baez said:

The only way I know how to do that is quote the question; you can click on a comment and quote it.

Like this?

Yes, like this. :slight_smile:

sarahzrf (Apr 08 2020 at 18:21):

okay, interesting, it looks like the definition of an ω-complete normed cone is equivalent to, like... "the norm preserves and lifts countable colimits (sups)" (in the senses here https://mathoverflow.net/a/293199/149197)

John Baez (Apr 08 2020 at 18:22):

Hmm, Sarah's comment makes me realize that maybe Prakash's "$\omega$-continuity" is just preservation of countable colimits (in the poset given by the cone).

John Baez (Apr 08 2020 at 18:22):

Is that right, Prakash? I threw in the word "just" just to show I'm a category theorist.

John Baez (Apr 08 2020 at 18:23):

Hey, Sarah re-edited her comment to make mine look dumber. :slight_smile:

John Baez (Apr 08 2020 at 18:23):

Now it looks like I'm just saying exactly what she said.

Prakash Panangaden (Apr 08 2020 at 18:23):

John Baez said:

Hmm, Sarah's comment makes me realize that maybe Prakash's "$\omega$-continuity" is just preservation of countable colimits (in the poset given by the cone).

Exactly. But that is always what $\omega$-continuity meant even in "domain" theory.

John Baez (Apr 08 2020 at 18:24):

Cool. I never learned domain theory.

Paolo Perrone (Apr 08 2020 at 18:24):

Is there an obvious reason for which we can restrict to countable sequences as opposed to directed nets in this case?

John Baez (Apr 08 2020 at 18:25):

I don't know, maybe Prakash knows but I'd guess restricting to countable sequences weakens the concept.

Oscar Cunningham (Apr 08 2020 at 18:25):

Everything has a real number norm, so how would you have an uncountable increasing sequence?

John Baez (Apr 08 2020 at 18:26):

Here's a related question: any poset gets an "order topology". How is $\omega$-continuity related to continuity in the order topology?

Prakash Panangaden (Apr 08 2020 at 18:26):

Paolo Perrone said:

Is there an obvious reason for which we can restrict to countable sequences as opposed to directed nets in this case?

I think the norm controls that.

Tobias Fritz (Apr 08 2020 at 18:26):

Paolo Perrone said:

Is there an obvious reason for which we can restrict to countable sequences as opposed to directed nets in this case?

Or put differently, what breaks if you require your cones to be more generally bounded directed complete and the maps to preserve all directed suprema?

John Baez (Apr 08 2020 at 18:28):

@Oscar Cunningham - you probably can't have an uncountable increasing "sequence" of real numbers, whatever that means, but you can have an uncountable net of them, like the net of numbers [0,1] indexed by the set [0,1].

Prakash Panangaden (Apr 08 2020 at 18:28):

Tobias Fritz said:

Paolo Perrone said:

Is there an obvious reason for which we can restrict to countable sequences as opposed to directed nets in this case?

Or put differently, what breaks if you require your cones to be more generally bounded directed complete and the maps to be preserve directed suprema?

I am not sure. We should think about that. If everything is countable it does not matter but there are uncountable directed sets whose limits you cannot get from sequences.

Prakash Panangaden (Apr 08 2020 at 18:30):

The interval [0,1] is the sup of the sup of the uncountable set of finite subsets of [0,1] ordered by inclusion but no sequence can get you there.

John Baez (Apr 08 2020 at 18:30):

I thought the concept of "normal state" involved some kind of countable continuity but the Wikipedia definition uses general nets:

John Baez (Apr 08 2020 at 18:30):

https://en.wikipedia.org/wiki/State_(functional_analysis)#Normal_states

Tobias Fritz (Apr 08 2020 at 18:30):

Prakash Panangaden said:

Tobias Fritz said:

Paolo Perrone said:

Is there an obvious reason for which we can restrict to countable sequences as opposed to directed nets in this case?

Or put differently, what breaks if you require your cones to be more generally bounded directed complete and the maps to be preserve directed suprema?

I am not sure. We should think about that. If everything is countable it does not matter but there are uncountable directed sets whose limits you cannot get from sequences.

Ok thanks. One thing that at least still works is that integration as a functional on L^oo(\mu) indeed preserves all bounded directed suprema.

But the subtlety here is that this would not be true without quotienting by null sets, i.e. integration does generally not preserve directed suprema. That's one way in which L^oo(mu) is nice

John Baez (Apr 08 2020 at 18:31):

A state \tau is called normal, iff for every monotone, increasing net H_{\alpha } of operators with least upper bound H, \tau (H_{\alpha })\; converges to \tau (H).

John Baez (Apr 08 2020 at 18:32):

Combining Tobias' remark with the Wikipedia definition I just spouted, I guess integration on L^infinity[0,1] is a normal state.

Paolo Perrone (Apr 08 2020 at 18:33):

I particularly like the duality of L^1 and L^oo cones, that's quite beautiful. Prakash, you mentioned that it was already known somewhere, right? Do you know where it was introduced?

John Baez (Apr 08 2020 at 18:33):

(Here I'm thinking of L^infinity[0,1] as a von Neumann algebra so the Wikipedia definition applies.)

Prakash Panangaden (Apr 08 2020 at 18:33):

Is there some way of copying this discussion to a document that I can stare at? There are lots of interesting remarks.

John Baez (Apr 08 2020 at 18:33):

I don't know - some Zulip expert would know.

Paolo Perrone (Apr 08 2020 at 18:33):

Prakash Panangaden said:

Is there some way of copying this discussion to a document that I can stare at? There are lots of interesting remarks.

I don't know, but this thread is here to stay :)

John Baez (Apr 08 2020 at 18:34):

@Paolo Perrone - I think Prakash & Co. sort of invented this duality of L^1 and L^infinity cones, but I think it has precursors elsewhere.

Prakash Panangaden (Apr 08 2020 at 18:34):

Paolo Perrone said:

I particularly like the duality of L^1 and L^oo cones, that's quite beautiful. Prakash, you mentioned that it was already known somewhere, right? Do you know where it was introduced?

I don't know anyone who said it like that but the people who were doing representations of C* algebra must have known it implicitly; else it is hard to understand why they would talk about normal states.

Tobias Fritz (Apr 08 2020 at 18:35):

John Baez said:

Combining Tobias' remark with the Wikipedia definition I just spouted, I guess integration on L^infinity[0,1] is a normal state.

Yes, right. The normal states are given by Prakash's cone M_ub (I think). In von Neumann algebra terms, the Radon-Nikodym derivatives of measures in M_ub are exactly the "density operators" associated to the normal states

John Baez (Apr 08 2020 at 18:35):

Right, Prakash - I was just gonna tell Paolo that analysts take a C*-algebra like L^infinity, whose dual is "too big", and trim it down using this concept of "normal state" I was just talking about.

John Baez (Apr 08 2020 at 18:36):

I forget what M_ub is.

Paolo Perrone (Apr 08 2020 at 18:36):

Oh, I see. Makes sense. Thanks!

John Baez (Apr 08 2020 at 18:36):

Okay, I guess I remember what M_ub is.

Prakash Panangaden (Apr 08 2020 at 18:37):

Is Arthur in this chat?

John Baez (Apr 08 2020 at 18:37):

I never got quite deep enough into C*-algebras to see why "normal states" are so great, but I guess it's precisely to eliminate things like the "weird" elements of the dual of L^infinity, and the "weird" states on the algebra of bounded linear operators, like the ones that vanish on all compact operators.

Paolo Perrone (Apr 08 2020 at 18:38):

I have a question again. Is Rad_1 (and/or Rad_oo) an opfibration over Meas?

John Baez (Apr 08 2020 at 18:40):

Hmm, I think only certain special measurable functions get to lift to morphisms in Rad_1 or Rad_infinity.

Prakash Panangaden (Apr 08 2020 at 18:40):

sarahzrf said:

(basically i'm wondering how much of this is property as opposed to structure)

I would love to understand this remark better.

Prakash Panangaden (Apr 08 2020 at 18:41):

Paolo Perrone said:

I have a question again. Is Rad_1 (and/or Rad_oo) an opfibration over Meas?

Can you remind me of the definition?

John Baez (Apr 08 2020 at 18:45):

An opfibration is roughly a functor F: C -> D such that every morphism in D "lifts" in a really nice way to one in C.

Prakash Panangaden (Apr 08 2020 at 18:45):

It looks like the rate of comments has slowed down. Thanks for all your great comments. There is much to think about.

John Baez (Apr 08 2020 at 18:46):

I was suggesting that Rad_1 -> Meas is not an opfibration because there are lots of morphisms that don't lift at all.

Prakash Panangaden (Apr 08 2020 at 18:46):

I need to leave now but I will check this link periodically.

John Baez (Apr 08 2020 at 18:46):

Okay, see ya!

Paolo Perrone (Apr 08 2020 at 18:47):

Ok, thank you so much for the talk! I hope to see you here in the future.

sarahzrf (Apr 08 2020 at 18:50):

John Baez said:

Now it looks like I'm just saying exactly what she said.

not quite—i'm talking about ω-completeness itself :)

sarahzrf (Apr 08 2020 at 18:51):

the property of the cone rather than of maps

sarahzrf (Apr 08 2020 at 18:52):

also oops i meant specifically of shape ω, not just "countable"

sarahzrf (Apr 08 2020 at 18:52):

i almost wrote that, too, but then i misread...

John Baez (Apr 08 2020 at 18:53):

Paolo Perrone said:

John Baez said:

I was suggesting that Rad_1 -> Meas is not an opfibration because there are lots of morphisms that don't lift at all.

Why not?
(To avoid confusion: by Meas I mean measurable spaces, not measure spaces.)

sarahzrf (Apr 08 2020 at 18:55):

man, i should learn about lawvere metric spaces

John Baez (Apr 08 2020 at 18:56):

John Baez said:

Paolo Perrone said:

John Baez said:

I was suggesting that Rad_1 -> Meas is not an opfibration because there are lots of morphisms that don't lift at all.

Why not?
(To avoid confusion: by Meas I mean measurable spaces, not measure spaces.)

Look at the definition of Rad_1 and Rad_infinity. Both these are categories where morphisms are measurable maps with an extra property. So these are non-full subcategories of Meas. So their inclusions into Meas are not cofibrations.

sarahzrf (Apr 08 2020 at 18:56):

i'm looking over these definitions of cones & whatnot and all i see is, like, lax monoidal functors and stuff :sweat_smile:

sarahzrf (Apr 08 2020 at 18:56):

wait, oh my god

Paolo Perrone (Apr 08 2020 at 19:04):

John Baez said:

John Baez said:

Paolo Perrone said:

John Baez said:

I was suggesting that Rad_1 -> Meas is not an opfibration because there are lots of morphisms that don't lift at all.

Why not?
(To avoid confusion: by Meas I mean measurable spaces, not measure spaces.)

Look at the definition of Rad_1 and Rad_infinity. Both these are categories where morphisms are measurable maps with an extra property. So these are non-full subcategories of Meas. So their inclusions into Meas are not cofibrations.

I think the objects have an extra structure, namely the measure. (Or am I getting confused?)

sarahzrf (Apr 08 2020 at 19:13):

okay i just confused myself

sarahzrf (Apr 08 2020 at 19:20):

@Prakash Panangaden is the multiplication on cones supposed to satisfy r(sv) = (rs)v?

sarahzrf (Apr 08 2020 at 19:21):

that doesnt seem to be mentioned anywhere? i think i was taking it for granted

christopher andrew upshaw (Apr 08 2020 at 19:27):

sarahzrf said:

Prakash Panangaden is the multiplication on cones supposed to satisfy r(sv) = (rs)v?

For s \in N that follows from distributing over addition, I think? r(3v) = r(v+v+v) = rv + rv + rv = r*3(v) ? Though maybe I am wrong about that first step?

sarahzrf (Apr 08 2020 at 19:28):

that sounds right, yeah

sarahzrf (Apr 08 2020 at 19:28):

actually no hold on

sarahzrf (Apr 08 2020 at 19:28):

yeah we also need (r + s)v = rv + sv

sarahzrf (Apr 08 2020 at 19:28):

for that to work

sarahzrf (Apr 08 2020 at 19:29):

which... doesnt seem to be mentioned...

sarahzrf (Apr 08 2020 at 19:29):

it does say "an action" of R^{≤0}, but since that was ambiguous i pulled up the paper and it mentions neither axiom :?

sarahzrf (Apr 08 2020 at 19:31):

i think ill just assume they were intended until i get a response

sarahzrf (Apr 08 2020 at 19:38):

anyway....... so the poset R⁺ w/ the standard ordering is a monoidal category under multiplication. if my assumption above was right, then we should have that the action of R⁺ on any given cone makes it an R⁺-actegory. then the lax-equivariant maps are the ones with rf(v) ≤ f(rv) for all r, v

sarahzrf (Apr 08 2020 at 19:41):

hmm... that does include all of the linear ones

sarahzrf (Apr 08 2020 at 19:42):

perhaps i am hunting a little too hard for optics here.

sarahzrf (Apr 08 2020 at 19:42):

:)

sarahzrf (Apr 08 2020 at 19:49):

thinking about how i want normed spaces to be spaces over R⁺, and then bounded maps are sort of like ones that make the triangle commute, except that actually it's a square and it doesn't commute it's just lax, so i don't even know why im trying to do this

John Baez (Apr 08 2020 at 22:11):

Paolo Perrone said:

I think the objects have an extra structure, namely the measure. (Or am I getting confused?)

You're right, sorry.

Let's see. We want to see if $p: \mathsf{Rad}_1 \to \mathsf{Meas}$ an opfibration, where $\mathsf{Meas}$ is the category of measurable space and measurable maps. I take a morphism $f : X \to Y$ in $\mathsf{Meas}$. I pick an object $X' \in \mathsf{Rad}_1$ and try to lift $f$ to some morphism $f' : X' \to Y'$.

John Baez (Apr 08 2020 at 22:14):

I'll just try to do this in some reasonably canonical way. The object $Y' \in \mathsf{Rad}_1$ should be a measure space over $Y$, so I'll push the measure on $X'$ forward along the measurable map $f$. Then we get a measure-preserving map of measure space and I think that's our $f' : X' \to Y'$ in $\mathsf{Rad}$.

John Baez (Apr 08 2020 at 22:15):

I'm too lazy to check to see if this is a "cocartesian lift", as required for an opfibration.

John Baez (Apr 08 2020 at 22:15):

But I've done the heavy lifting here.

Paolo Perrone (Apr 09 2020 at 01:21):

Yep, that's kind of what I'm hoping for! I guess the rest is taken care of by the fact that the UB and almost continuity relations form preorders. Nice! :)

John Baez (Apr 09 2020 at 05:36):

sarahzrf said:

i think ill just assume they were intended until i get a response

Yes, cones had better obey the laws (r + s)v = rv + sv and r(sv) = (rs)v; we couldn't do much without those laws.

David Michael Roberts (Apr 09 2020 at 23:53):

For what it's worth, in the YouTube video description is says the talk was on the 1st April, so presumably a copy and paste from @John Baez 's talk?

Joe Moeller (Apr 10 2020 at 17:24):

Thanks, I just fixed that.

John Baez (Apr 10 2020 at 17:42):

It was my fault.

David Michael Roberts (Apr 11 2020 at 05:34):

@Joe Moeller thanks!

Alexander Kurz (Apr 23 2020 at 03:50):

I missed the talk, unfortunately. Is there a recording? I browsed through the thread above, but didnt see a link ...

John Baez (Apr 23 2020 at 18:17):

All talks at the ACT@UCR seminar are recorded and available here, along with links to slides and/or relevant papers, a schedule of forthcoming talks, etc:

https://johncarlosbaez.wordpress.com/2020/03/24/actucr-seminar/

This page has a link to our YouTube channel as well as links to videos of individual talks.

Alexander Kurz (Apr 24 2020 at 05:34):

@Rongmin Lu Thanks

Arthur Parzygnat (Apr 26 2020 at 10:43):

Prakash Panangaden said:

Is Arthur in this chat?

I am now! But I noticed you might not be checking this very often anymore? Thanks @Tobias Fritz for asking my question in the chat and sorry for not being here to be a part of that discussion. In regards to the question, yes, I was thinking that maybe the inner product equation Panangaden wrote is Bayes' rule (expressed without using denominators) specialized to the case when alpha is deterministic. The associated map on the function algebras is the pullback. When you work with Markov kernels, you use integration to extend the notion of pullback, but the equation itself does not change in form. Since the time I asked the question in your talk, I had time to read the original paper, but I haven't yet put the details together in my head since I'm less familiar with the ω-continuous aspects. When the underlying sets are finite, the formula agrees with Bayesian inversion in the way I understand it (since L_\infty is isomorphic to L_1 and Bayesian inversion is well-defined a.e.). Since alpha is deterministic, this is just a disintegration.

As for the infinite-dimensional setting, we know the story on the space level with measurable functions (that's the usual disintegration theory for standard Borel spaces). When we go to the function spaces, we have many options, all the L_p spaces for example. But only one of them has a multiplication operation to make sense of the Bayes condition (just so we have the same definition, I'm using the diagram (6.4) in https://arxiv.org/pdf/2001.08375.pdf adapted to the algebraic setting where the 2 to 1 map, depicted as a trivalent bullet, read from top to bottom is multiplication---see also (8.26)) and that's L_\infty with the probability measure viewed as a positive linear functional on the algebra. But Panangaden does not go this route and instead uses an (L_p, L_q) duality, and I'm not 100% sure why. Here's my guess, based on a first reading of "Dahlqvist Danos Garnier Kammar - Bayesian inversion by ω-complete cone duality" https://hal.archives-ouvertes.fr/hal-01429656/document. I believe that their Proposition 17 combined with Theorem 19 say that the inner product equation Panangaden wrote is indeed Bayes' equation. It seems that this approach of using ω-complete cones has the benefit of not requiring assumptions on the underlying measure spaces (standard Borel for example). Is that the main point?

Jade Master (May 06 2020 at 18:26):

(deleted)

Paolo Perrone (Sep 01 2020 at 18:48):

The third talk in this series is about to happen! Thread here: https://categorytheory.zulipchat.com/#narrow/stream/229457-MIT-Categories.20Seminar/topic/September.203.3A.20Prakash.20Panangaden's.20talk