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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: Re: Fuzzy theory or probability theory? 
Message-ID: <1994Dec6.173831.11033@decan.gate.net>
Organization: Decision Analytics, Inc.
Date: Tue, 6 Dec 1994 17:38:31 GMT
References: <94Nov29.133132edt.774@neuron.ai.toronto.edu> <1994Dec5.190928.3981@newsserver.rrzn.uni-hannover.de>
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mackw@bytex.com wrote:

: In article <94Nov29.133132edt.774@neuron.ai.toronto.edu>, 
: <radford@cs.toronto.edu> writes:
: > To take an actual real-world situation, consider the problem of a 
: > police officer who has interviewed a witness to a crime.  The witness
: > described the perpetrator as "tall".  How should we formalize how the
: > officer should handle this information?
: > 
: > I think it is clear that what the police officer needs here is a 
: > likelihood function - for any actual height (plus any other 
: characteristics
: > that might influence subjective assessment of height), the officer
: > needs to know the PROBABILITY that the witness would describe the
: > such a perpetrator as "tall".

(( cuts ))

: 1) Correct me if I'm wrong, but I don't believe that any probability 
: measure of what heights are considered tall exists or are in use by 
: police departments.  This would seem to contradict the statement that
: the police officer needs this information to understand the description 
: of "tall."

The fact is every competent speaker of the language already has this
information.  While there are no statistical tables that I know of,
I think we all know the answer to the following question: what is the
probability that a randomly chosen, competent speaker of the language,
would use the word "tall" to describe a perpetrator who is 5'6" in
height.  The answer is "pretty close to 0".  On the other hand,
for a perpetrator who is 6'6" in height, the answer is "pretty close
to 1".  For heights in-between, the probabilities are in-between.  
The fact that, as speakers of the language, we "know" these things,
helps to explain the ease with which fuzzy practitioners can draw
these membership functions, seemingly relying only on subjective feel.

: 2) In reference to a different post, we need to remember that english and 
: fuzzy logic equations are two different languages that we must translate 
                            ^^^^^^^^^^^^^^^^^^^^^^^
: between.  Thus the english language statement "tall and not tall" is not 
: equivalent to the fuzzy language statement "tall AND NOT tall."  

Then you concede that in the (min-max) fuzzy set theory, the conjunctive
operator AND and the negation operator NOT do not correspond with
natural semantic law.

: The 
: correct translation of the fuzzy statement is "tall to a degree and not 
: tall to another degree."  

Although you have jumped into the meta-language, where the sense in
which there is no contradiction is perfectly clear, you still _use_
the fuzzy-logic rendition "tall AND NOT tall" to correspond to
a _single_ utterance "tall and not tall" in the object language. 
It is this object-language utterance which violates the law of
(non-)contradiction, notwithstanding the fuzziness of the term in
question, "tall".  If your fuzzy-logic formalism is to capture meaning
in the object language, then the fuzzy-logic "tall AND NOT tall"
also should yield the fuzzy-logic parallel of what in the object
language is the absurdity required by the law of (non-)contradiction.
Can you have a formalism which violates this law?  Sure.  Does it 
then correspond to natural-language semantics?  The obvious answer is
no.  But fuzzy logic sets out to model the fuzziness and imprecision
inherent in natural language.  If in so doing it violates semantic
laws even more basic -- after all, the meta-language in which the
theory is couched is ordinary bivalent mathematical logic, where 
violation of LNC would render it impossible to prove even the
simplest theorem in the fuzzy logic being advanced -- then it seems
to me that the theory should in principle be modified to conform to
the very law without which even the meta-language would fail.  In
_Fuzziness_and_Probability_ I show how that can be done.

: Literal, word for word translation, from fuzzy 
: to english is no more valid than for any other language translation and 

No, no.  Please don't elevate the thing doing the modelling (fuzzy
logic) to the level of the thing being modelled (natural-language
semantics).

: can easily lead to confusion when we forget we are doing a translation 
: step.

The confusion arises from going the wrong way.  You should go from
english to fuzzy, and retain semantic law in so doing, not the other
way 'round.

: 3) It is quite easy to restate a problem from terms of "membership" to 
: terms of "probability."  An item's membership of a certain characteristic 
: is related to the probablity that this characteristic will cause 
: something to happen or not happen.  Restating the problem from one domain 
: to another merely shows the problem statement exists in both domains; it 
: does not imply which solution domain (fuzzy logic or probability) should 
: be used.

After the fresh semantics provided by Zadeh, it really doesn't matter
if fuzzy logic (after some rework) could be shown to derive from probability.
In fact, both fields would benefit from closer association with the
other.

: 4) Fuzzy logic does not concern itself very much with how the membership 
: set equation is determined, probability is extremely concerned with 
: determining this shape.  Fuzzy logic assumes that if a membership set 
: definition is reasonably close, then the result will also be reasonably 
: close.  It may also be noted that sampling methods also do not provide a 
: precise membership set definition and usually includes an approximation 
: of the likelihood that the definition is wrong.  Furthermore, the 
: practice of curve fitting is itself a fuzzy operation; normal 
: distributions do not exist in the real world (this is *not* saying they 
: are not useful, I'm just trying to say they are approximations).

Fuzzy logic is supposed to be crisp, even if it treats of a fuzzy
subject matter.  I hope you are not saying that the fuzziness of the 
subject matter is reason to be conceptually sloppy in the fundamentals
of the theory.

(( cuts ))

: In summary, if probability information is available or easily attainable, 
: use it to determine membership sets.  The advantage of fuzzy logic is in 
: cases where you do not want to obtain this data to this level of 
: precision, you do not have to give up and say the problem is impossible 
: to solve.  This why products are appearing using fuzzy logic which were 
: not made using probability.  They  probably were possible using 
: probability, it just wasn't worth the effort.

The joining together of probability and fuzziness concepts is motivated
not by estimation problems concerning the latter, but by a need to
achieve conceptual clarity.  Here is an excerpt from
_Fuzziness_and_Probability_ which addresses this issue:
   
   "The likelihood school takes the position that
   the likelihood function may be regarded as a 
   direct characterization of the uncertainty regarding 
   [an unknown parameter of a statistical model].
   If only a sufficiently powerful
   likelihood calculus could be developed, as easy of 
   interpretation and manipulation as the probability 
   calculus, we could avoid the indirect circumlocutions
   of the classical school, also avoid when
   necessary the insistent subjectiveness of the 
   Bayes school, while securing, one hopes, all the 
   advantages of a direct characterization of uncertainty.
   A fair amount of effort has been expended
   in attempting to develop such a calculus.  The 
   calculus so far developed fails, however, in limiting cases,
   throwing up difficulties of interpretation,
   and encounters severe difficulties in the
   elimination of nuisance parameters.
   (Kalbfleisch and Sprott, 1970; Plante, 1971; Edwards, 1972;
   Basu, 1973; Hill, 1973).
   In the same situations, the Bayesian probability calculus
   faces no similar difficulty. If only the likelihood
   calculus could be suitably modified and extended,
   the potential exists for an approach to
   inference that could conceivably combine the manipulative
   advantages of the Bayesian approach, 
   while meeting the genuine concerns of the classical school 
   regarding the insistent subjectiveness of the 
   Bayes school. Just such is proposed in this essay 
   \*(EM a possibility calculus that modifies and extends 
   the likelihood calculus in ways that meets the 
   current difficulties. 
   
   The redevelopment of the likelihood calculus
   which is proposed stems from the conception of the 
   likelihood function as characterizing a fuzzy set. 
   It is this conception that forges the link between
   statistical inference
   and fuzzy set theory.  Some of the foundational 
   difficulties of fuzzy set theory become
   amenable to resolution through the need to bring 
   the concepts of probability and fuzziness into the 
   intimate relationship implied by the conception of 
   likelihood as fuzzy set.  Thus both the theory of statistical inference
   and the theory of fuzzy sets appear to be strengthened
   by the unification.

   The absolute likelihood ...
   nevertheless is well defined and may be accorded a 
   separate existence and even a separate interpretation.
   That is, it provides a measure of truth for
   any simple hypothesis that may be considered. 
   Since $ L ( omega ) $ is defined equal to the probability
   of the result assuming the hypothesis $ theta ~=~ omega $,
   it follows that $ L ( omega ) $
   ranges on the interval  $ [ 0 , 1 ] $.  In this it may be 
   construed as a membership function, and so, formally
   at any rate, it defines a fuzzy subset of $ OMEGA $,
   the parameter space.  I would contend
   that this fact has a useful interpretation, not 
   just a formal one, with positive implications both 
   for fuzzy set theory and for a likelihood calculus.
   The interpretation is simple: the fuzzy subset
   in question may be construed as that corresponding
   to the predicate "explains the data x".  The
   universe of discourse to which the fuzzy subset 
   belongs is the set of probability distributions 
   having as their domain the sample space.  It is not 
   at all stretching the notion of the fuzzy set to 
   admit within its ranks the class of models that 
   explain observed data.  The essential criterion for 
   classification as a fuzzy set is met: there are 
   degrees to which members may belong to the "set"
   in question.  That is to say, the predicate "explains
   the data x" is a fuzzy descriptor of some
   unknown model, or model parameter, in the same way 
   that the predicate "is tall" may be used as a 
   fuzzy descriptor of some (to a listener) unknown 
   height value.  One may expect similar meta-semantic 
   considerations to govern the choice of combination 
   rules -- the development of a calculus -- to deal 
   with the two situations.  I want to return to this 
   question later on, as it represents the \fIraison 
   d'etre\fR of the development."

: Wayne Mack
:  

Cheers!
S.F.Thomas
