In the interest of focussing the discussion, I have snipped some
of the points and counter-points raised by Mr. Sewell.
Martin Sewell wrote:
[...]
> "How can we best interpret fuzzy theory using traditional mathematics 'from
> within'?"
>
> The fuzzy membership function could be interpreted as a probability density
> function (pdf). The fuzzy set then becomes a set with a probability
> distribution attached and the degree of membership of the set becomes a
> probability.
This is not quite correct. A degree of membership needn't be a probability
distribution, since there is no requirement that the membership function
be integrable. However, a (normalized) membership function can indeed be
interpreted as a likelihood function -- that is, a function of the
conditioning variable in a conditional distribution. E.g., if p(x|y) is
a conditional probability, then g(y) = p(x|y) for a fixed x, is a like-
lihood function.
> "But I've been told that I mustn't confuse probability with degree of
> membership. In fuzzy theory if we say that a man is tall with a degree of
> membership of 0.8, we don't mean that he is tall 80% of the time."
There is a misconception here. There is no need to restrict
probability to "random" processes, as implied by the "tall 80% of the time"
statement. Probability applies to the same set of propositions that
classical logic applies to; this result is due to R.T. Cox and others.
I can see that part of the motivation for fuzzy logic is to represent
uncertainty which is not associated with random events. However, it is
not necessary to invent a new logic for that purpose; probability is
quite sufficient.
> "What's inherently wrong with fuzzy theory?"
>
> Fuzzy theory forces continuous data into discrete sets, which necessarily
> effects a loss of information.
Maybe so, but I don't think this problem is at all important.
The important problems are these:
(1) Fuzzy logic doesn't handle dependent variables correctly.
(2) You cannot bet on (i.e., allocate resources according to)
degrees of membership.
Regards,
Robert Dodier
-- ``There lives somewhere a hookah-smoking genius locked in a Chinese Room filled with ideographs and untranslatable requests put through slots from the outside. For this he is paid fifty dollars an evening.'' --Ron Bloom
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