Re: category theory and probability theory

Re: category theory and probability theory

[F W Lawvere]

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which also was discussed in an unpublished 1962 paper of mine, is that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a monoidal functor was exploited in the unpublished doctoral thesis here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.
> 
> 
> 

Re: category theory and probability theory

[boerger]

	I like to add some references concerning applications of category 
theory to measure theory:

  - Fred Linton´s 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 

Re: category theory and probability theory

[Amilcar Sernadas]

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. <cotton@ensae.fr>
To: categories@mta.ca <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.
>

category theory and probability theory

[jean-pierre-C.]

  
  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.