Re: Lawvere on probability distributions as a monad
Dear Jeremy Gibbons, Strangely enough, since I have worked "extensively" for years on Lawvere distributions, I cannot help you locate the preprint in question. But I can answer some of the questions you pose. For that, I will quote our book, Marta Bunge and Jonathon Funk, Singular Coverings of Toposes, LNM 1890, Springer 2006 and references therein. It can be downloaded from the web. There is indeed a (Kock-Zoberlein) monad M (with unit the "Dirac delta" on the 2-category Top_S of Grothendieck toposes (actually toposes E bounded over a base topos S), called the "symmetric monad", such that the S-points of M(E), for E a topos, is the category of distributions on E in the sense of Lawvere, that is, the category of cocontinuous functors E-->S and natural transformations between them. The existence of the symmetric topos was established by myself (Bunge 1995) using forcing topologies, and given an "algebraic" construction in (Bunge-Carboni, 1995), including the KZ-monad structure. Of course we study (Bunge-Fuk 2006) the M-algebras and several other matters. Concerning probability distributions, there is a classifying monad too. A probability distribution on a topos is a distribution that preserves the terminal object. It is shown (Bunge-Funk 2006, Proposition 8.2.6) that, for any S-topos E there is a subtopos T(E) of M(E) that classifies the probability distributions on E. This means that for any E there is an equivalence between the category of probability distributions on E and that geometric morphisms E--> T(E) over S. The Dirac delta E--> M(E) is an S-essential geometric morphism so given by a 3-tuple of S-adjoints delta_! adj delta^* adj delta_*. T(E) is "simply" the subtopos of M(E) given by the least topology forcing delta(1) --> 1 to be an isomorphism. There is also a construction of T(E) in terms of sites (Proposition 8.2.9, op.cit.) On the other hand, I know nothing about your comment
(and so probabilistic computations can be captured as a "computational effect").
so that, if such a preprint exists, I too would like to have it. I Will ask Lawvere when (or if) I see him in a month unless he replies himself to your query. With best wishes, Marta Bunge ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************
From: jeremy.gibbons@cs.ox.ac.uk Subject: categories: Re: Lawvere on probability distributions as a monad Date: Mon, 13 Jun 2011 22:28:31 +0100 To: categories@mta.ca
Dear all,
I sent the following request a few months ago, looking for a "preprint" of Lawvere's from 1962. I'm not the only one interested - I got five replies, off-list, of the form "If you find it, could I have a copy too, please?"
I thought I would try one last time. My paper that cites this has been accepted for publication, and I'm doing due diligence by trying my best to track down original sources!
Does anyone know where I can find a copy?
Thanks again, Jeremy
On 22 Feb 2011, at 14:08, Jeremy Gibbons wrote:
I wonder if you fine categorists could help me track down an old preprint?
Many people have written about probability distributions forming a monad (and so probabilistic computations can be captured as a "computational effect"). The reference trail goes back to
Michele Giry, "A Categorical Approach to Probability Theory", LNM 915:68-85, 1981
and thence to
F W Lawvere, "The Category of Probabilistic Mappings", preprint, 1962
I have Giry's paper, but can find no trace on the web of Lawvere's preprint. Does anyone know where I might find a copy? Might someone even have a copy that they would be prepared to scan?
Thanks, Jeremy
Jeremy.Gibbons@cs.ox.ac.uk Oxford University Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. +44 1865 283521 http://www.cs.ox.ac.uk/people/jeremy.gibbons/
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Marta, I too am interested in seeing Bill's preprint, but is it indeed about the distributions in the sense of the symmetric topos? (Let us agree on calling those Lawvere distributions.) Although the formal analogies between Lawvere distributions and ordinary, measure theoretic, distributions are very profound, it seems to me it would be misleading to talk as if they were a categorical generalization of the ordinary ones. You and Funk have shown that, restricting to localic toposes, the localic reflection of the symmetric topos is the lower powerlocale. But that is not adequate in itself for describing probabilistic features. For that we need the valuation locale. I don't see that the symmetric topos has that extra measure-theoretic nature that is absent from the lower powerlocale and has to be added in the valuation locale. Roughly speaking, for the valuation locale, the lower powerlocale, and the symmetric topos, the extensive and intensive quantities take their values in the space R of, respectively, lower reals (0 to infinity inclusive), subsets of 1 (R = Sierpinski), and sets (R = object classifier). Your result about localic reflection then restricts to the fact that Sierpinski is the localic reflection of the object classifier. OK, I've indulged myself a bit with a spiel on something I've been thinking about, but there's a more straightforward question about the content of Bill's preprint. Is it (1) Lawvere distributions, (2) something more genuinely probabilistic, or (3) an account of how the former give us the latter? All the best, Steve. --- The valuation locale V(X) has as its points the continuous valuations on the frame of X, functions m from the frame to the lower reals from 0 to infinity, such that m is Scott continuous, m(0) = 0 and m has the modular law m(U)+m(V)=m(U\/V)+m(U/\V). It does not completely capture measure theory, but those aspects that can be derived from the measures on the opens of spaces. It relates to the probabilistic powerdomain of computer science. See - Vickers "A localic theory of lower and upper integrals" Coquand and Spitters "Integrals and valuations" Marta Bunge wrote:
Dear Jeremy Gibbons,
Strangely enough, since I have worked "extensively" for years on Lawvere distributions, I cannot help you locate the preprint in question. But I can answer some of the questions you pose. For that, I will quote our book, Marta Bunge and Jonathon Funk, Singular Coverings of Toposes, LNM 1890, Springer 2006 and references therein. It can be downloaded from the web.
There is indeed a (Kock-Zoberlein) monad M (with unit the "Dirac delta" on the 2-category Top_S of Grothendieck toposes (actually toposes E bounded over a base topos S), called the "symmetric monad", such that the S-points of M(E), for E a topos, is the category of distributions on E in the sense of Lawvere, that is, the category of cocontinuous functors E-->S and natural transformations between them. The existence of the symmetric topos was established by myself (Bunge 1995) using forcing topologies, and given an "algebraic" construction in (Bunge-Carboni, 1995), including the KZ-monad structure. Of course we study (Bunge-Fuk 2006) the M-algebras and several other matters.
Concerning probability distributions, there is a classifying monad too. A probability distribution on a topos is a distribution that preserves the terminal object. It is shown (Bunge-Funk 2006, Proposition 8.2.6) that, for any S-topos E there is a subtopos T(E) of M(E) that classifies the probability distributions on E. This means that for any E there is an equivalence between the category of probability distributions on E and that geometric morphisms E--> T(E) over S. The Dirac delta E--> M(E) is an S-essential geometric morphism so given by a 3-tuple of S-adjoints delta_! adj delta^* adj delta_*. T(E) is "simply" the subtopos of M(E) given by the least topology forcing delta(1) --> 1 to be an isomorphism. There is also a construction of T(E) in terms of sites (Proposition 8.2.9, op.cit.)
On the other hand, I know nothing about your comment
(and so probabilistic computations can be captured as a "computational effect").
so that, if such a preprint exists, I too would like to have it. I Will ask Lawvere when (or if) I see him in a month unless he replies himself to your query.
With best wishes, Marta Bunge
************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
As someone who for many years had a copy of that preprint, and may yet have a copy still, buried in the mess a rabid Dean made of the records in my office (by packing them all willy nilly in boxes and stacking those boxes into several six-foot tall columns in front of two bookshelves that are still holding books (!)), let me at least outline what I remember of it. [I might hope that one of the several folks I shipped copies of it to in years past might still have -- and share -- such a copy.] The work itself Bill developed in the early sixties, while both a grad student at Columbia and an employee of Litton Industries. The heart of it is a category -- rather Kleisli-category-like now, in retrospect, the way it's built, though the very notion of Kleisli category had not yet broken through the categorical consciousness -- whose objects, as I recall, were pairs made up of a set X and a boolean sigma-algebra A of subsets of X, while the maps from one such object (X, A) to another (Y, B) were those functions f: X --> prob(B) (from X to the set prob(B) of probability measures on B) for which, separately in each variable, each f(x, =): B --> R is a probability measure on B (yes, already said), each f(-, b): X --> R is an A-measurable real-valued function on X. For the composition of such an f with g: (Y, B) --> (Z, C), note that each f(x, =): B --> R is a probability measure on B and that each g(-, c): Y --> R is a B-measurable real-valued function on Y; so we may, for each x in X and c in C, form the integral (over B) {\Integral}_B g(-, c) d(f(x, =)) of the real-valued function g(-, c) on B w/ resp. to the measure f(x, =) and call that real number (g.f)(x, c) . The slogans "Associativity = Fubini" and "Identity = Dirac Delta" outline how one sees this is a category. In the absence of the actual purple hexographed spirit document, I'm unable to reconstruct much more. But I hope this helps. And if Bill is tuning in to this thread, I'd be grateful if you could fine-tune what I've said, Bill, wherever I've gotten things off-pitch or out of key, and perhaps amplify what I've left too quiet. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Erratum -- end of sentence after {\Integral} display in prior version -- corrected. Sorry. -- F. | To the Editor: if possible, suppress the earlier version and use this one instead, with these top 10 lines excised. Thanks, -- Fred ------ Original Message ------ Received: Mon, 20 Jun 2011 03:37:49 PM EDT From: "Fred E.J. Linton" <fejlinton@usa.net> To: <categories@mta.ca> Cc: Steve Vickers <s.j.vickers@cs.bham.ac.uk>, <martabunge@hotmail.com> Subject: Re: categories: Re: Lawvere on probability distributions as a monad As someone who for many years had a copy of that preprint, and may yet have a copy still, buried in the mess a rabid Dean made of the records in my office (by packing them all willy nilly in boxes and stacking those boxes into several six-foot tall columns in front of two bookshelves that are still holding books (!)), let me at least outline what I remember of it. [I might hope that one of the several folks I shipped copies of it to in years past might still have -- and share -- such a copy.] The work itself Bill developed in the early sixties, while both a grad student at Columbia and an employee of Litton Industries. The heart of it is a category -- rather Kleisli-category-like now, in retrospect, the way it's built, though the very notion of Kleisli category had not yet broken through the categorical consciousness -- whose objects, as I recall, were pairs made up of a set X and a boolean sigma-algebra A of subsets of X, while the maps from one such object (X, A) to another (Y, B) were those functions f: X --> prob(B) (from X to the set prob(B) of probability measures on B) for which, separately in each variable, each f(x, =): B --> R is a probability measure on B (yes, already said), each f(-, b): X --> R is an A-measurable real-valued function on X. For the composition of such an f with g: (Y, B) --> (Z, C), note that each f(x, =): B --> R is a probability measure on B and that each g(-, c): Y --> R is a B-measurable real-valued function on Y; so we may, for each x in X and c in C, form the integral (over B) {\Integral}_B g(-, c) d(f(x, =)) of the real-valued function g(-, c) on Y w/ resp. to the measure f(x, =) on B and call that real number (g.f)(x, c) . The slogans "Associativity = Fubini" and "Identity = Dirac Delta" outline how one sees this is a category. In the absence of the actual purple hexographed spirit document, I'm unable to reconstruct much more. But I hope this helps. And if Bill is tuning in to this thread, I'd be grateful if you could fine-tune what I've said, Bill, wherever I've gotten things off-pitch or out of key, and perhaps amplify what I've left too quiet. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
Fred E.J. Linton -
Marta Bunge -
Steve Vickers