Re: Fuzzy logic compared to probability

Darren J Wilkinson (D.J.Wilkinson@durham.ac.uk)
Mon, 18 Mar 1996 16:33:42 +0100


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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?

: 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?!

: 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.

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