Re: Fuzzy logic compared to probability

Darren J Wilkinson (D.J.Wilkinson@durham.ac.uk)
Tue, 5 Mar 1996 20:21:59 +0100


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

[ An extremely long posting about the latest attempt of the computer ]
[ science and AI communities to avoid having to understand the ]
[ foundations of statistical inference. Much of this posting is, ]
[ IMVHO, ill-concieved, but, unlike Thomas, I do not have time to ]
[ address all points in detail. However, I will focus on one aspect ]
[ which illustrates the lack of understanding of basic statistical ]
[ knowledge the AI community is lacking. ]

: I take the (perhaps simplistic) view that modelling is fundamentally
: about counting and classifying within an assumed morphology
: (objects and measureable attributes thereof, concerning which discourse
: proceeds) for the phenomenon in question. Classification requires
: measurement, and, ipso facto, observation. Counting may lead
: to a probability hypothesis. But probabilities may never ever
: be directly *observed* as a singular event or observation...
: measured outcomes, yes (eg. "heads" or
: "tails" on the toss of a coin, for a simple nominal-scale example,
: but not P(Heads)=0.5). Singular events can never literally be
: repeated, our universe being in perpetual motion. But what repeats
: is the morphology we mentally construct around phenomena as
: we seek to bring order (counting and classifying) to our observations.
: It is this notion of morphology that bridges the gap between
: frequency and subjective notions of probability. It is the notion of
: morphology that provides the link between separate performances
: of the same (frequentist) experiment as somehow being connected.

Only a frequentist thinks of "repeated observations". 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 the
notion that beliefs over a collection remain invariant under an
arbitrary permutation of elements of the collection. Such a notion leads
to a representation for the collection which separates belief into
belief about the "underlying true value", and "individual vatiation".
Weaker forms of symmetry often lead to similar representations. It is
these representations which allow understanding of underlying processes,
and allow future prediction.

There are two fundamental misconceptions at the heart of fuzzy theory.
The first is that uncertainty can be adequately understood without the
notion of probability. Probability theory exists precisely as a language
for the understanding of uncertainty. 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....

.. There is no way, however, in which the individual can avoid the burden
of responsibility for his own evaluations. The key cannot be found that
will unlock the enchanted garden wherein, among the fairy-rings and the
shrubs of magic wands, beneath the trees laden with monads and noumena,
blossom forth the flowers of PROBABILITAS REALIS. With these fabulous
blooms safely in our button-holes we would be spared the necessity of
forming opinions, and the heavy loads we bear upon our necks would be
rendered superflous once and for all.

Bruno de Finetti
Theory of Probability, Vol 2

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--
Darren Wilkinson  -  E-mail: d.j.wilkinson@durham.ac.uk 
<a href=http://fourier.dur.ac.uk:8000/djw.html>WWW page</a>