Date: Jun 9, 2006 1:37 PM
Author: Christopher J. Henrich
Subject: Re: bayesians versus frequentists

In article <e6bu1o01189@drn.newsguy.com>, Daryl McCullough
<stevendaryl3016@yahoo.com> wrote:

> dave says...
> 
> >The frequentist view fits into the currently fashionable set-theoretic
> >foundational framework for mathematics better than the Bayesian view,
> >and hence, the pure mathematicians favor it.
> 
> I don't see that at all. I would think that a pure mathematician would
> reject frequentism as nonrigorous. The probability of X, according to
> frequentism, is the limit (number of instances of X)/(number of trials)
> as the number of trials goes to infinity. But mathematically speaking,
> there is no reason for this ratio to have a limit.
> 
> The pure mathematician is more likely to define probabilities axiomatically:
> A probability distribution on a set E is a partial function from the power
> set of E to [0,1] such that blah, blah, blah. (Where blah blah includes
> countable additivity, etc.)
> 
To the pure mathematician, this is a satisfactory definition of a topic
in pure mathematics, within which there are plenty of problems to
solve, theorems to prove, and so on.  The debate starts when people try
to apply the mathematics to non-mathematical problems. 

For some time I was of the opinion that applications of probability
theory tended to be of two kinds, and the difference between
"frequentists" and "Bayesians" was that they were interested in
different applications. The different kinds of applications are:

1. Statistical mechanics.  Here, it is quite impossible to track
trajectories of a dynamical system with 10^24 degrees of freedom, but
if we replace the individual trajectories by a suitably chosen measure
on configuration space, we get something that is mathematically
tractable and useful.

2. Insurance. Here, we cannot predict which building will have a fire,
but in a large enough ensmble of buildings we can predict, closely
enough, how many will have fires.

3. Estimation theory.  Here, probabilites are quantitative expressions
of imperfect knowledge.

(Wait.  There are three different kinds of applications: quantum
mechanics; statistical mechanics... I mean, /four/ kinds: quantum
mechanics, statistical mechanics, insurance, estimation, and fanatical
devotion to the Pope.)

In my list, insurance is pretty clearly a frequentist application, and
estimation is a Bayesian application.  I am not sure whether
statistical machanics is frequentist or Bayesian. 

To have a debate about frequentism versus Bayesianism, the debaters
must agree that the question, "What is the meaning of proability
theory?" has one answer which is both all-inclusive and all-exclusive. 
It has to be inclusive of all possible uses of probability theory, and
it has to be exclusive of all alternative meanings.

-- 
Chris Henrich
http://www.mathinteract.com
God just doesn't fit inside a single religion.