Re: Lawvere on probability distributions as a monad

["Fred E.J. Linton" <fejlinton@usa.net>]

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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