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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Fuzzy logic compared to probability
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Date: Mon, 11 Mar 1996 00:25:05 GMT
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Darren J Wilkinson (D.J.Wilkinson@durham.ac.uk) wrote:
: -----BEGIN PGP SIGNED MESSAGE-----

: S. F. Thomas (sthomas@decan.com) wrote:

: : 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?

: Indeed it is, but the infereences one would draw are IMHO, necessarily 
: subjective. Suppose that you would like to know the probability of 
: throwing a head with this coin. You toss it 5 times, and it comes down 
: heads each time. Would the inferences you make about the "true" 
: probability of a head be the same, whether or not the coin was given to 
: you by the pope, who swore it to be fair, or by a crooked gambler, who 
: also claimed it to be fair. I think not. This is, ultimately, why those 
: who wish "the data to speak for themselves" are chasing the end of a 
: rainbow. In statistics, the "true" behaviour, partly obscured by noise, 
: gradually emerges as more and more data become available. The rate at 
: which the data should "swamp" your prior beliefs is necessarily 
: subjective. What would _you_ estimate the probabilty of a head to be in 
: the above example, and why?

I happen to agree that Bayesians are quite right to pay attention
to subjective prior belief in situations where the formal data
are limited, or where the subjective prior belief may be deemed
reliable for any variety of reasons specific to the situation
at hand.  In the example of coin-tossing, I have no difficulty
with a procedure -- as for deciding which team goes first
in a game -- which presumes that a coin has equal chances coming
up heads or tails if fairly tossed.  I am being deliberately
circular here to make a point.  The coin could be fair, as would
be revealed by simple inspection (does it indeed have two 
differently marked sides? is its weight uniformly distributed?),
yet the coin *toss* may be unfair, as could be the case if the
person making the toss is well practiced at making it come
up whichever side he wishes.  No problem.  One person makes
the toss, and the other person calls it... while the coin
is in the air.  For such a procedure, I, and countless sportsmen
every day, would be satisfied a priori that the chance of
winning was 50:50, ie. the coin (tossing procedure, actually)
was fair.  And indeed you are right to suggest
that if a captain lost the toss five times in a row, using
such a validated *procedure*, it should be put down to bad luck
rather than an unfair *coin*.  If it went on for an extended
period of time (with the same coin and tosser) one would
tend to get suspicious, and should -- like a schoolboy contemplating
outliers in the data -- look for hidden error, or deliberate
cheating, whether the tosser was the pope, or a known crook. 
One should never throw away common sense, certainly not in
the name of an inferential paradigm, Bayesian or otherwise.

I disagree with the Bayesian approach when it is claimed
that subjective prior belief must *always* be injected,
whatever the inference and/or decision problem at hand.
I also disagree with the use of Bayes theorem that treats
uncertainty in universals (that regarding unknown model
parameters) symmetrically with occurrence uncertainty in 
particulars.  There are some other disagreements as well
(Thomas, 1995) which I won't elaborate here.

: : The essential stratagem revealed -- making of a bug, a feature.
: : I do not deny the importance of subjective belief in certain
: : situations.

: Ha! Which situations, and do you _subjectively_ decide which?! That isn't 
: very "objective", is it?!

I have given some general considerations above -- namely
when subjective prior knowledge is available, and the 
formal data are few, or when prior knowledge is deemed reliable
for whatever reason that may be applicable to a choice problem
at hand.  Our modelling of the world proceeds
at (at least) two quite distinct levels.  At the first, we 
construct a morphology (sometimes only implicitly) 
for the phenomenon being modelled.
That requires insight, aesthetics, judgement, intuition
and other mental faculties that defy prior logic.  
Call those subjective if you wish.  That I do not
argue.  Einstein's insight was to bring the phenomenon 
of observation into the consideration of the phenomenon 
of motion, thus changing the absolutist morphology 
advanced by Newton.
It is only when we have a morphology that we have a
frame within which to make our observations and to order
them.  It is at this second level of making and ordering
observations that the issues at hand are essentially
objective.  And even if the morphology were arrived
at by an essentially subjective process, the external
reality sought to be modelled at the second level,
remains quite independent of whatever one's prior 
expectations might be.

: : But to take this ounce of truth, and to make of it
: : the whole inferential meal, is, again, sidestepping the essential
: : problem of inference, which is to characterize what *the data*
: : say.

: The data alone say _nothing_. You could take a perfectly fair coin, toss 
: it a million times, and it _could_ come up heads every time. It is only 
                                                               ^^^^^^^^^^
: _your_ belief about the underlying probability that matters. It is only 
  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
: in the context of your quantified belief that you can think about how 
: observations should modify your belief. 

I disagree, for reasons already stated.  

: --
: Darren Wilkinson  -  E-mail: d.j.wilkinson@durham.ac.uk 

Regards,
S. F. Thomas
