Date: Tue, 20 Oct 1998 08:13:01 +0200 (DFT)
From: "jean-pierre-C."
Subject: categories: category theory and probability theory
Bonjour. I am a statistician and I should be interested in a categorical
framework for probability and statistical theory. Does anyone know
references (books, articles, websites...) about applications of categories
and functors to probability or even measure theory ? Thank you.
Very truly yours,
Jean-Pierre Cotton.
From: "Amilcar Sernadas"
Subject: Re: categories: category theory and probability theory
Date: Wed, 21 Oct 1998 10:06:42 +0100
We are working on a related problem. It seems that it is necessary to work
with
a relaxed notion of category, namely where the compostion of
f:a->b and g:b->c is not always defined. You should look at relaxed
notions of category such as composition graphs, paracategories,
precategories
and the like.
On our own preliminary results look at the working paper
P. Mateus, A. Sernadas and C. Sernadas. Combining Probabilistic Automata:
Categorial Characterization. Research Report, April 1998. Presented at the
FIREworks Meeting, Magdeburg, May 15-16, 1998
that you can fetch from
http://www.cs.math.ist.utl.pt/s84.www/cs/pmat.html
Amilcar Sernadas
-----Original Message-----
From: jean-pierre-C.
To: categories@mta.ca
Date: Quarta-feira, 21 de Outubro de 1998 0:19
Subject: categories: category theory and probability theory
>
> Bonjour. I am a statistician and I should be interested in a categorical
>framework for probability and statistical theory. Does anyone know
>references (books, articles, websites...) about applications of categories
>and functors to probability or even measure theory ? Thank you.
>
> Very truly yours,
>
> Jean-Pierre Cotton.
>
Date: Sun, 25 Oct 1998 13:18:33 -0500 (EST)
From: F W Lawvere
Subject: categories: Re: category theory and probability theory
In reply to the query of Jean-Pierre Cotton, I would like to mention the
following:
In Springer LNM 915 (1982), an article by Michele Giry develops some
aspects of "A categorical approach to probability theory".The key idea,
which also was discussed in an unpublished 1962 paper of mine, is that
random maps between spaces are just maps in a category of convex spaces
between "simplices". There is a natural (semi) metric on the homs which
permits measuring the failure of diagrams to commute precisely, suggesting
statistical criteria. To make full use of the monoidal closed structure,as
well as to account for convex constraints on random maps,it seems
promising to consider also nonsimplices. (Noncategories is usually not a
good idea). The central observation that the metrizing process is actually
a monoidal functor was exploited in the unpublished doctoral thesis here
at Buffalo by X-Q Meng a few years ago in order to clarify statistical
decision procedures and stochastic processes as diagrams in a basic
convexity category. She can be reached at : meng@lmc.edu
Best wishes to those interested in pursuing this topic!
Bill Lawvere
*******************************************************************************
F. William Lawvere Mathematics Dept. SUNY
wlawvere@acsu.buffalo.edu 106 Diefendorf Hall
716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA
*******************************************************************************
On Tue, 20 Oct 1998, jean-pierre-C. wrote:
>
> Bonjour. I am a statistician and I should be interested in a categorical
> framework for probability and statistical theory. Does anyone know
> references (books, articles, websites...) about applications of categories
> and functors to probability or even measure theory ? Thank you.
>
> Very truly yours,
>
> Jean-Pierre Cotton.
>
>
>
From: boerger
Date: Tue, 27 Oct 1998 15:52:52 +0100
Subject: categories: Re: category theory and probability theory
I like to add some references concerning applications of category
theory to measure theory:
- Fred Lintonīs thesis and his paper "Functorial measure theory" in
the Irvine Proceedings, Thompson , Washington D.C., !966.
- Various papers by Mike Wendt, in particular his thesis on direct
integrals of Hilbert spaces
- My still unpublished long paper "Vector integration by universal
properties" and its simplified version in the Bremen Proceedings,
Heldermann, Berlin, 1991.
Kind regards
Reinhard