Re: Bayesian & Frequentist Probability Theory

[Bill Jefferys <bill@warthog.as.utexas.edu>]

At 11:22 PM +0000 3/9/02, Charles Francis wrote:

#In article <bill-1260D7.18571306022002@newshost.cc.utexas.edu>, Bill
#Jefferys <bill@warthog.as.utexas.edu> writes
#
#>At 8:01 AM +0000 2/2/02, Charles Francis wrote:
#
#>In my experience, many of the objections to Bayesianism are due to
#>misconceptions about what it actually is. If your objection to
#>Bayesianism is what you wrote above, you should do some reading and
#>rethink your position.
#
#I have taken this on board. My interest lies in what it means to say
#that a mathematical structure is a model of physics, an in particular
#what probability theory means. This impacts on the questions:
#
#a) whether it is legitimate to modify probability theory to produce
#quantum mechanics
#b) If so, whether the justification for doing so necessarily comes from
#a Bayesian perspective

Probability theory is probability theory. There is no need to modify 
probability theory from any perspective in order to do quantum 
mechanics. Bayesianism uses standard, unmodified probability theory. 
Bayesian interpretations of QM use standard, unmodified probability 
theory.

There are folks who propose to solve QM problems with negative 
probabilities, complex probabilities, etc., (e.g., Stanley Gudder, Saul 
Yousef); some may call these Bayesian interpretations, but I would not 
call them Bayesian. Such approaches are also not necessary and in my 
opinion they confuse more than they illuminate.

#c) whether such a modification of probability theory qualifies as an
#interpretation of quantum mechanics.
#
#My introduction to the Bayesian/Frequentist debate was from the
#perspective of those questions, and with hindsight, I think it was
#misleading.
#
#The issues I am concerned with are quite different from those regarding
#the Bayesian or Frequentist statistical analysis of data. It is a very
#long time since I did this sort of statistical analysis, and when I did
#I don't remember anyone talking about the Bayesian-Frequentist question.
#However having looked at it I have little doubt that the Bayesian
#methodology is preferable, and gives a more accurate reflection of what
#we ought to believe regarding the parameters of a model underlying a
#data sample, though, as you say, both methods have validity.
#
#However this seems to be a modern meaning of the adjective Bayesian,
#having more to do with the use and significance of Bayes theorem in
#identifying belief intervals than it has to do with the use or abuse (as
#the case may be) of Bayesian reasoning in understanding either
#probability theory or quantum mechanics.
#
#Bayesian reasoning is quite different, and concerns the use of
#subjective probability measures in decision making processes even in the
#absence of a meaningful data sample. 

??? There is a subjective component to Bayesianism, of course: there is 
a prior, and there is a utility function when doing decision theory. But 
I don't know any Bayesian who would advocate making decisions without 
data. You need a prior, of course, but the prior is a means to an end. 
Ditto the utility function. But ultimately the data rule.

#This isn't really probability
#theory at all, but has more in common with many valued logic and
#questions that we would probably address with fuzzy logic these days.
#
#The position I have objected to is that because probability theory is
#ultimately Bayesian and subjective it is legitimate to modify
#probability theory in order to fit the data, and that this constitutes
#an interpretation of quantum mechanics. This position seems to confuse a
#probability theory with ideas taken from Bayesian reasoning.
#
#I would agree, someone who proposes this is confused.
#
#On the other hand I have concluded that quantum theory is an exotic
#probability theory, and that it is not only legitimate but also correct
#to modify the laws of probability theory to describe properties of a
#particular class of physical model. But I do not think I do so on
#specifically Bayesian grounds.

It's not necessary to do this. The issue is to condition on what you 
know. If you do this, then standard probability theory is all you need.

Bill

-- 
Bill Jefferys/Department of Astronomy/University of Texas/Austin, TX 78712
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