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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Fuzzy logic compared to probability
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Date: Thu, 7 Mar 1996 17:08:05 GMT
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Herman Rubin (hrubin@b.stat.purdue.edu) wrote:
: In article <DnsyEI.6u5@decan.com>, S. F. Thomas <sthomas@decan.com> wrote:
: >Darren J Wilkinson (D.J.Wilkinson@durham.ac.uk) wrote:

: 			.....................

: >: Only a frequentist thinks of "repeated observations". 

: >I would have thought that Bayesians also were (in part) frequentist,
: >since the likelihood function, which derives from frequentist 
: >probability models, is central to the updating of Bayesian prior
: >belief.

: This is a complete misunderstanding.  Any attempt to come up with a
: theory based upon action, and not based upon something as crude as
: data description, requires that the assumptions come from the "mind".
: Frequentism only comes in from the inference standpoint.
  ^^^^^^^^^^^^^^^^^^^^^^^^^^
So you concede my point, which was only that Bayesians are,
indeed have to be, *in part* frequentist.  I do not disagree
that "personal" elements enter into any sort of decision-making,
first with respect to characterizing *fact* -- about which
in general there is uncertainty, and often personal experience
which ought to be factored in, and to which Bayesians are
in principle quite right to pay attention -- and second with
respect to sorting out *preference* which is a quintessentially
personal matter.  As to preferences, there is no doubt that
such concepts as utility, level of desirability, degree of
aversion, etc., as used in the contemplation of the consequences
of available options, have psychological origin.  And I
certainly do not attempt to claim otherwise.  But as to 
observable and measureable fact, which is a concern of
frequentist models, one may make subjective estimates, but
such estimates do not have an intrinsically psychological
origin, any more than a subjective estimation of a psycho-physical
attribute (eg. loudness, pitch, sweetness) is intrinsically
psychological in origin.  I'm making a fine distinction, but
the sense of it should be clear.  It is that the external
reality, whether of the loudness of the noise, or *the data*
observed in a frequentist experiment, exists quite independently
of any particular decision-maker who seeks to make use of
such observations or reality in a decision-making context.
It is in contemplation of *the data* that Bayesians too
must be frequentist.

: >: For a Bayesian, the 
: >: link is provided by consideration of symmetry and invariance, the 
: >: strongest form of which is given a name: "Exchangeability".

: This is what I have been calling "rash Bayesian" behavior for decades.
: There are two versions of invariance; changing the units, or changing
: the presentation of the problem, would not change a Bayesian's results.
: But if changing the units came up with a FORMALLY identical problem,
: the results, in general, SHOULD be different.  Symmetry likewise, 
: unless one's assumptions are symmetric.

This is not clear to me, but as I'm neither Bayesian, nor
rash, I may not be in need of the lesson.  Still, though, I
wish I understood it.

: 			...................

:                                           Subjective prior belief,
: >no matter how sophisticated the mathematics, or how clever the
: >trotting out of such concepts of "exchangeability", remains an
: >artful dodge where the central problem of inference is concerned,
: >which is, how to characterize what *the data* say about some unknown
: >probability distribution of interest.  Throwing in a statistician's --
: >or a "user's" -- prior belief into the mix, is still, in my opinion,
: >sidestepping the real question.  And arguing for the *necessity*
: >of so doing, not to mention the *goodness* of so doing, is the
: >Bayesian equivalent of turning a bug into a feature.

: You and too many others are overly concerned with belief.  If, instead,
: one starts with the idea of self-consistent behavior, which is much
: weaker than one would think, it is still strong enough to be only
: compatible with a weighted sum of the probabilities of outcomes in
: the various states of nature, the weights depending on the action. 
: If these weights are called the product of loss and prior, this gets
: the more customary formulation.  There is no operational way of
                                   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
: separating loss from prior.
  ^^^^^^^^^^^^^^^^^^^^^^^^^^

I disagree.  You think so because you refuse to step outside
of the Bayesian paradigm.  As I have indicated in another post: 

  "If you step out of the Bayesian paradigm for a minute, and contemplate
   the possibilities opened up by an extended likelihood calculus that is 
   just as powerful as the probability calculus in the manipulations it 
   allows, I think the challenge (major, IMO) of formalizing prior beliefs 
   in a manner consistent with, eg. the total ordering axiom, would be seen
   to be the unnecessary trap that it is, rather than the virtue
   that is so often claimed."

: 			...................

: >: The other is that such 
: >: understandings do not have to be fundamentally subjective. Frequentist 
: >: statisticians have been trying to be "objective" for decades, and the 
: >: literature is littered with examples of it's abject failure. I will end 
: >                                               ^^^^^^^^^^^^^^
: >: this post with a quote by a man who understood uncerainty better than 
: >: anyone had ever done before....

: >You make assertions, or rather dogmatic statements of Bayesian
: >doctrine.  But you do not make rational arguments to which one could
: >respond.  Surely, not even a Bayesian would deny that observing
: >heads or tails on the toss of a coin is essentially an "objective"
: >procedure, whatever one's prior belief might be as to which
: >might turn up?

: The OBSERVATION is objective.  The action to take should not be the same
: for all observers.  If it were, Michelson and Morley would have come up
: with special relativity as soon as they made their observations.  Newton
: would have come up with the wave theory of light.  Archimedes and Euclid
: would have used variables.  The structure of "elementary particles" would
: have been deduced years before.
: 			

No disagreement from me on the basic point that preferences are
subjective.  If you like vanilla, it does not prevent me from
preferring chocolate.  But I dislike your examples, because they
confuse external reality (fact) with preferences.  The modeling
of the real world is not a solipsistic endeavor, and the 
benchmarks for measuring how well we do at it are far removed
from mere assertions of personal preference ... although in 
looking for "order" in the universe, we do exhibit an aesthetic
sense that may vary from observer to observer; but that is
not the point.

 			..................
: -- 
: Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
: hrubin@stat.purdue.edu	 Phone: (317)494-6054	FAX: (317)494-0558

Regards,
S. F. Thomas
