Re: Fuzzy logic compared to probability

Shiva P. Gautam (spg@uhura.neoucom.edu)
Fri, 8 Mar 1996 22:11:14 +0100


Darren J Wilkinson (D.J.Wilkinson@durham.ac.uk) wrote:
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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".

I did not know that that the concept of exchangeability was developed
in the context of Bayesian statistics.

: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.

IMHO, frequentists approach is probability can not be subjective not "do
not have to subjective. Reading these threads it looks like fuzzicist
approach is essentially it does not have to be subjective approach in all
situations. "The theorem" which says that probability is the only way
which can model uncertainty is again IMVHO is scary and counter intuitive.
(I have to admit that not all solution of a problem confirm to intuition).
This thread is about understanding the relationship between probability and
fuzziness, and if it is kept that way there is no danger of digression. If
you honor my subjective opinion, then there are situations and evidences
of failure of frequentist approach but not to the extent you mentioned. I
have seen Bayesians writing frequentist text books. Bayesian use
likelihood a frequentist notion, and frequentist use subjective approach
in many thins for example which methods to use, what distributions to
assume etc. But again I might be roaming in a different world.

:I will end
: this post with a quote by a man who understood uncertainty 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
Indeed it is a great quotation. But it reminds me of S. F. Thomas who puts
his points somehow in a similar fashion (choice of words, metaphors,
lucidity, eloquence etc.) which you seem to dislike.

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

Shiva