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Added a reference of Robert Furber, Bart Jacobs at Giry monad.
I noticed that Giry monad used to jump into some details without really saying what’s going on first. I have now
added a minimum of an Idea-section;
split the “Outline” section into a Definition-section (and tried to make it read more like an actual definition) and a “Properties – Algebras”-section (which also deserves some streamlining, but I haven’t touched this);
touched the formatting of some of the references.
Experts who care might want to polish this entry a bit more.
I’ve added a reference to Tom Avery’s paper arxiv:1410.4432 and renamed the entry to ’Giry monad’ from ’Giry’s monad’.
Some of the literature on the Giry monad seems to be behind a paywall. Can anyone tell me whether the weak topology on the space $P(X)$ of Borel probability measures is the same as the topology induced by the Prokhorov metric? I have just added the latter as an example to further examples at Polish space, and would like to add this material plus a reference to Giry monad as well, but would like to check up on that point first.
Yes, at least for a complete separable metric space (that is to say Polish). Reference: Dudley, real analysis and probability.
Perfect; thank you Daniel. Is there a Theorem or page number where the statement is made?
complete answer: The statement holds actually even for any separable metric space. Reference: Dudley, Real Analysis and Probability, 2002, Theorem 11.3.3 page 395. Note that Dudley defines Prokhorov metric not according to normal terminology but introduces it calling it simply $\rho$. Anyway the cited theorem states the equivalence of all these notions of convergence, to roughly summarize the theorem: (almost) all reasonable notions of convergence of laws coincide on separable metric spaces (If you add even complete, I could not think of any reasonable notion that would not be equivalent to weak convergence).
Thank you again, Daniel. This is very helpful indeed.
I’m curious about the work by Voevodsky mentioned in this page. Unfortunately, I don’t speak Russian so I can’t watch the Moscow lecture. Does anyone know of any English-language paper, notes, or recorded lecture on this work?
Hi Evan. I looked around at the time that was added, but I never found anything.
Perhaps try contacting Zoran Skoda who wrote the part about Voevodsky.
Somebody at the Miami lecture must have taken notes. Could we try contacting somebody there? Looking at the faculty list, it is not obvious who might have invited him/known him, though.
Perhaps a good start is contacting Daniel Grayson?
You need to choose Markdown+Itex, so
Dan Grayson has now made it available here
Very nice. Thanks everyone for your help!
Hi, Someone just pointed out that, earlier this month, some claims were added by Kirk Sturtz about “super convex sets” and his recent arxiv preprint. I don’t entirely understand the preprint or the claims. Not sure how best to handle this here.
Thanks for the alert!
The additions by Sturtz are in rev 31, rev 32 and some reformatting of references in rev 33.
Haven’t looked into the actual content, but it does seem strange that rev 31 effectively changes the previous content without further notice into the terminology apparently introduced in Sturtz’s article. At the very least it seems the original content should have been kept, and instead a remark be added that an alternative has been suggested.
Not sure how best to handle this here.
Since we haven’t seen Sturtz here, while you are a regular, and if, as you seem to say, you are familiar with the subject, have looked into the edits and the article, and find that it does not make sense, then the way to handle this is to roll back to before these edits, hence to rev 30.
Sturtz had made earlier edits to the page, including items which the new material addresses. It would be good to have some expert input. The page is probably due an overhaul anyway.
If one looks at some of the earlier revisions which David mentions, it seems that Kirk has only either added material or modified bits which already referred to his work. Thus perhaps a reasonable step would be to send an email and invite him to join this discussion and perhaps give an overview of his contributions to the page thus far?
Here is the bit I cut (for now).
The results of Doberkat can be generalized to the Giry monad $G$ on all measurable spaces by using the factorization of the Giry monad through the category of super convex spaces $\mathbf{SCvx}$, by viewing the Giry monad itself as a functor into that category. (A super convex space is similar to a convex space except the structure requires that if $\{\alpha_i\}_{i=1}^{\infty}$ is any countable partition of unity (so the limit of the sum is one), then for any sequence of points in a super convex space $A$, the countable sum $\sum_{i=1}^{\infty} \alpha_i a_i$ is also an element of the space. The morphisms in the category preserve the countable affine sums. The right adjoint of that functor assigns to each convex space the measurable space, defined on the underlying set, with the initial $\sigma$-algebra generated by all the countably affine maps into the one point extension of the real line $\mathbb{R}_{\infty}$. The construction of the counit amounts to using the fact that the full subcategory consisting of the single object $\mathbb{R}_{\infty}$ is condense in $\mathbf{SCvx}$. This implies that every $\mathbb{R}_{\infty}$-generalized point of a super convex space $A$ is an evaluation map, at a unique point $a$ of $A$. Using this fact, given any arbitrary probability measure $P$ defined on $\Sigma A$, one takes the restriction of $P$, viewed as an operator $\mathbf{Meas}(\Sigma A, \mathbb{R}_{\infty}) \rightarrow \mathbb{R}_{\infty}$, mapping $f \mapsto \int_A f \, dP$, to the subset of countably affine maps, $\mathbf{SCvx}(A, \mathbb{R}_{\infty}) \rightarrow \mathbb{R}_{\infty}$. This restriction process yields an $\mathbb{R}_{\infty}$-generalized point in $\mathbf{SCvx}$ which is necessarily a unique point of $A$ since $\mathbb{R}_{\infty}$ is condense in $\mathbf{SCvx}$.
As an illustration, the half open interval $[0,\infty)$ is not a super convex space because one can take the countable partition of one given by $\{\frac{1}{2^i}\}_{i=1}^{\infty}$, and a set of points $\{i 2^i\}_{i=1}^{\infty}$ in $[0,\infty)$ so that the countably infinite sum $\{\sum_{i=1}^{\infty}\frac{i 2^i}{2^i}\}_{i=1}^{\infty}$ does not exist. This shows that $[0,\infty)$ is not a super convex space and explains why there is no barycenter map for this space. (Consider the half-Cauchy distribution.) On the other hand, the open unit interval, $(0,1)$ is a super convex space and does have a barycenter. This illustrates that compactness is not a requirement for a barycenter map to exist, only the property of being a super convex space is necessary. (But it does tie in a sequential completeness condition - thereby making a connection with the topological viewpoint.)
The category of super convex spaces is equivalent to the category of Giry algebras.
@Sam, if you edit #26 and select the Markdown+Itex option and resubmit, it will be properly formatted.
A note about my personal confusion about these developments from Sturtz.
As far as I understand, a superconvex space is like a convex space except also allowing infinite formal sums weighted by convergent sequences of reals. (This is reasonable, and ties to Avery/Sturtz’s codensity characterizations, which should be mentioned properly somewhere.) But if Giry-algebras are supposed to be the same as plain superconvex sets, then I don’t see where the additional structure of a sigma-algebra comes from. I can see how to make up a sigma-algebra on an arbitrary superconvex set, but I don’t see why this is the only consistent sigma-algebra. Maybe the sigma-algebra is actually supposed to be given as extra data, but this is not mentioned in the latest arxiv paper by Sturtz.
@DavidRoberts, thanks, re #27.
I don’t want to appear like a bookie but would it be accepted if I put a footnote there for this bit of private communication?
I think that would be good.
I feel like some part is missing from this story. Why was it attached as an appendix to this unrelated report?
I have added to the text pointer to the reference in question.
Can we give poor Michèle Giry a hyperlink?
I suggest to move the section “History” from the top to the bottom of the entry. This is distractive miscellenania barely on topic, which the reader should not need to skip through to find out what a Giry monad is in the first place.
Can we do something about this abomination of a non-reference?
Doberkat has a longer article on Eilenberg-Moore algebras of the Giry monad as item 5 here. (Unfortunately, the monograph ‘Stochastic Relations: Foundations for Markov Transition Systems’ doesn’t appear to be available.)
now that link is broken, it is useless. Why not give the name? Was item 5 the named paper or something else? What if the numbering had changed?
Edit: the monograph is available, as a book for purchase: https://books.google.com.au/books/about/Stochastic_Relations.html?id=_OmI_xrYqawC&redir_esc=y, so this sentence makes one think it is nowhere available. I will edit the page to fix this, but not now. I will see if I can track anything down about the mysterious “item 5”, if no one else knows what it might be.
It’s been there right from the start rev 1.
OK, so “item 5” in this Internet Archive capture is
- E.-E. Doberkat, Ch. Schubert: Coalgebraic Logic Over General Measurable Spaces – A Survey. Math. Struct. Comp. Sci. 21 (2), 2011, 175 - 234 (pdf). We discuss in this survey the generalization of stochastic Kripke models for general modal logics through predicate liftings for functors over general measurable spaces. Results on expressivity are derived, and it is shown that selection arguments permit incorporating the discussion of bisimilarity, provided the underlying spaces are assumed to be Polish.
which doesn’t look right. Looking at those entries that mention ’Eilenberg’ or ’Giry’, it’s hard to tell which, if any, our article is meant to be describing.
This reference seems to have been added by Zoran in May 2013, which is over a year from the last Wayback machine capture, so it’s conceivable the items were renumbered, though Coalgebraic Logic Over General Measurable Spaces – A Survey is the best bet for what was meant. I may have a look tomorrow at the article to see if it is even relevant, if Zoran (or someone else) doesn’t pipe and correct me.
About Michèle Giry, I met her just once in Madame Ehresmann’s office in Amiens. I believe she became a secondary school teacher and then went into teacher training. She used to have a webpage in the University of Amiens but that link is now dead so I presume she has retired.
I remember discussing one Doberkat’s paper on stochastic stuff and Giry monads and generalizations with Roland Fridrich at the time, actually several years earlier. Later Doberkat had some new articles which touched the topic and I got hold of the book only later. Which was original viewed, I should look at time stamps in my old directories. I can not do in the very moment, but I will check this later, thanks.
40: according to vague memory it was this Dortmund preprint Characterizing the Eilenberg-Moore algebras for a monad of stochastic relations, listed at Giry monad. In any case it was not from a journal but a preprint of similar look and the content fits.
Dmitri Pavlov wrote at #35:
I feel like some part is missing from this story. Why was it attached as an appendix to this unrelated report?
Apparently this appendix was intended to provide a reasonable framework for the development of verification protocols in the arms control context. We also have Lawvere on video telling the story himself! I will add this link to the page.
I have taken the liberty of turning the footnote into an item in the list if references – because that’s what it is and how you are using it in the text. Then I replaced the unspecified “in a video comment” with actual pointer to that reference.
(Am relieved to hear that the Pentagon doesn’t disagree with the category of probabilistic mappings.)
Between #21 and #28, there was discussion of additions by Sturz. He’s now brought out a new paper on Giry algebras over measurable spaces. Any views?
With all this attention being paid to categorical probability theory now, surely some expert out there could help.
In the formula for the multiplication of the Giry monad (why is it called the “counit”?), what is P(X) in the integral? P(X) is only defined well below this formula. It looks like the formula should actually say G(X), not P(X).
The relevant change is Revision 23 by K. Sturtz on October 30, 2018.
I’ll guess what is meant is that you take the inverse images of measurable sets of $[0, 1]$, which gives you a $\sigma$-algebra on $G(X)$.
Corrections/Modifications. In the section on Algebras over the Giry monad: (1) Doberkat’s example is the free algebra on $\mathbf{2} =\{0,1\}$. It is rather useless as an example precisely because it is free. This leads to the second correction, (2) There seems to be a fundamental misunderstanding why one necessarily needs a concrete representation. I hope the added paragraph explains the necessity (concerning existence of non free algebras). (3) I updated my reference to my latest work - which gives the simplest most direct way to show the existence of non free algebras using the support of a probability measure (=added Lemma) which I wish I understood 10 years ago!)
Kirk Sturtz
In explaining why we are naturally led to super convex spaces I added the relatively short proof that every G-algebra specifies a super convex space and, likewise, that every map of G-algebras is a countably affine map.
Incidentally, I suspect that Doberkats category may have the Klesi category = category of Algebras because continuous map do not permit discrete spaces so everything is embeddable into a vector space which yields the algebras as $\mu$ = averages over all probability measures.
Kirk Sturtz
I added the reference to Ruben Van Belle’s article which can be viewed, in hindsight, as naturally suggesting the two constructions (dense and codense functors) which arise when considering super convex spaces. The explanation in the main article (on $G$-algebras) is the easiest way to motivate the constructions and explain why the countable set $\mathbb{N}$ is sufficient”.
Kirk Sturtz
I suggest that if you request an author hyperlink, as in
that you immediately go and create that author page (just a brief page: pointer to the author’s web presence plus the reference).
Because otherwise nobody will do, typically, in which case the broken link looks bad.
added publication data for this item:
I rewrote the section on Algebras. The main change is that I wanted to point out why there are no non-free algebras for the case of Polish spaces. That proof points out explicitly why we need to think of the Giry monad via superconvex spaces - every algebra and every morphism of algebras is necessarily a countably affine map. You can’t prove the fact that the category of algebras is equivalent to the Kleisi category of the Giry monad without that fact which is why Doberkat didn’t recognize that his description of the algebras were the free algebras.
Kirk Sturtz
Isn’t the closed unit square a non-free algebra?
Hi Kirk.
I do agree that superconvex spaces have been much overlooked in the past, and should be used more.
I still think that [0,1] x [0,1] is not free. The reason is that [0,1] x [0,1] is isomorphic to [0,1] in the category of Polish spaces (and measurable maps), but not in the category of Giry algebras. Said more precisely, given the componentwise algebra structure on [0,1] x [0,1] which you mentioned, we have that any isomorphism of measurable spaces [0,1] x [0,1] -> [0,1] is not a morphism of algebras. (Not even of convex or superconvex spaces).
The situation is similar for the category of groups: the group Z is free (over one generator), and as a set it is isomorphic to Z x Z, since they are both of countable cardinality. However, the group structures are not isomorphic (and the second one is not a free group).
To prove that [0,1] x [0,1] is not free as a convex space it suffices to show that its universal property fails. The universal property boils down to saying that every point is a unique convex combination of extreme points. Now the extreme points of [0,1] x [0,1] are the vertices of the square: (0,0), (0,1), (1,0), and (1,1). But now the point (1/2, 1/2) can be expressed both as 1/2 (0,0) + 1/2 (1,1) and as 1/2 (0,1) + 1/2 (1,0). (Geometrically, the square is not a simplex.)
Also, I think that in the next few days I should give a thorough makeover to the Giry monad article, a ton of research has been made since it was written.
Paolo, I finally understood your concern.
Given an arbitrary space $X$ which is of cardinality $c$, we know there exists a unique map $h:GX \rightarrow X$ but how do we know $h=\mu_Y$ for some space $Y$? The concern is with the homeomorphism $X \rightarrow G(2)$ preserving the (super)convex space structure. I need to think about this.
Added corrections for the section on Algebras for the Giry monad which address the error pointed out by Paolo Perrone which result in showing the explicit construction of a G-algebra on Polish Spaces which is not free. Paolo, if you see any improvements to the arguments please make changes as needed. The section on the Kleisi category is woefully lacking. (But I confess having no interest in that.)
I still need to add back the reference to Doberkats article in the noted section.
Kirk Sturtz
I added Doberkats discussion of barycenter maps which is how most people think about probability monads. Using the theory of barycenters requires some kind of compactness which is not a requirement of Polish spaces, however there are P-algebras which do satisfy that condition.
I also edited the proof of the Lemma concerning the fact that P(2)xP(2) is not isomorphic to P(3). By showing P(3) is a quotient of P(2)xP(2) that result is more obvious, and that proof generalized to the more general case quite easily.
Kirk Sturtz
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