Re: Lawvere on probability distributions as a monad

Re: Lawvere on probability distributions as a monad

[Fred E.J. Linton]

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






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Re: Lawvere on probability distributions as a monad

[Jeremy Gibbons]

Dear categorists,

I finally found a copy of Lawvere's 1962 preprint, thanks to the assistance of Kirk Sturtz. I'd like to thank you all for your help in locating it.

For the record, there is apparently a photocopy of it in Gian-Carlo Rota's papers in the American Institute of Mathematics library:

   http://www.aimath.org/library/library.cgi?database=reprints;mode=display;Title=;Author=lawvere;Codenumber=14044

Kirk kindly sent me a scanned copy of this photocopy (with Kirk's own annotations).

Cheers,
Jeremy

On 13 Jun 2011, at 22:28, Jeremy Gibbons wrote:

> 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
> 

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Re: Lawvere on probability distributions as a monad

[Fred E.J. Linton]

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




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Re: Lawvere on probability distributions as a monad

[Marta Bunge]

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/


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Re: Lawvere on probability distributions as a monad

[Jeremy Gibbons]

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/ ]

Lawvere on probability distributions as a monad

[Jeremy Gibbons]

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@comlab.ox.ac.uk
    Oxford University Computing Laboratory,    TEL: +44 1865 283508
    Wolfson Building, Parks Road,              FAX: +44 1865 283531
    Oxford OX1 3QD, UK.
    URL: http://www.comlab.ox.ac.uk/oucl/people/jeremy.gibbons.html





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