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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.
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