I have not followed this thread but our review on interpretations and
measurement of fuzzy membership functions may be of relevance.
T. Bilgiç and I.B. Turksen (1999) "Measurement of Membership Functions:
Theoretical and Empirical Work" Chapter 3 in D. Dubois and H. Prade
(eds)
Handbook of Fuzzy Sets and Systems Vol. 1, Fundamentals of Fuzzy Sets,
Kluwer, pp. 195-232.
http://www.ie.boun.edu.tr/~taner/publications/papers/membership.pdf
Taner
Kathryn Blackmond Laskey wrote:
>
> Scott,
>
> >> > First, remember that it's not the *probability* of being
> >> > tall or small. It's not a probability at all. It's something
> >> > else, sometimes called "possibility", which measures the
> >> > degree something is true (not its frequency, or even one's
> >> > belief that it's true).
> >> This is true, but allow me to remark that I haven't seen any better
> >> `definition' than ``it's something else''. No axiomatic foundations, such
> >> that you can never be sure whether it's your calculus or your algorithm
> >> that leads to bad results....
> >
> >Well, they do have a clear axiomatic foundation. I agree however
> >that the fuzzy types have not given a clear interpretation of what
> >possibility really *is*. What is this measure really measuring?
> >...
> >> > In a fuzzy set theory, the set of tall people
> >> > and the set of small people could well be not mutually
> >> > exclusive. I'm tall for a jockey, but pretty small for a
> >> > basketball player. It makes a difference what the sets
> >> > were constructed to represent.
> >...
> >So you think vagueness is "nothing more than incomplete
> >information"? It's easy to show that it has nothing to do with
> >incomplete information. I could have all the heights of every
> >single individual in the population down to the nanometer,
> >yet still not be sure whether someone deserves the appellation
> >of "tall". There are still borderline cases. Or did you mean to
> >say it is nothing more that incomplete *specification*? That's
> >the more common argument.
>
> Probability is appropriate for sets satisfying the "clarity test." That
> is, could a clairvoyant who knows the entire state of the world, past
> present and future, down to the wave function of every quark, unambiguously
> specify the value of the variable in question? For heights measured in
> centimeters, the answer is yes (leaving out quantum fuzziness, which is
> there but matters only in the fifteenth decimal place or so). For example,
> our clairvoyant can easily answer questions such as whether my son Robbie
> will be between 175 and 176 centimeters tall when he reaches his full adult
> height. Therefore, it is fully appropriate to use a probability density
> function on his adult height, (at least in the classical physics
> approximation where people have definite heights -- which will serve most
> of our modeling purposes just fine).
>
> However, as pointed out, even if we knew Robbie's adult height, we wouldn't
> know whether he will be tall or not. I agree with the fuzzy folks that
> there *is* something there that's important to capture. However, I've
> tried in vain to get a number of different people in the fuzzy community to
> tell me what a fuzzy membership actually means in operational terms. If
> I'm going to use something in a serious engineering application, as opposed
> to academic philosophizing, it is *very* useful to know what I'm doing in
> theory, even if I do put in plenty of engineering hacks. As my thesis
> advisor used to tell me, "First figure out what you would do if you could
> do it right, and then figure out how to approximate it." If I don't KNOW
> what the thing I'm trying to approximate with my engineering hacks would
> mean if I could do it right, I'm rather uncomfortable.
>
> For probability theory we have several competing ontologies that have clear
> operational meaning in the domains to which they apply. The most commonly
> cited are (1) propensities based on physical symmetries; (2) limiting
> frequencies of "random" events; (3) beliefs about uncertain phenomena. All
> of these give clear operational criteria for connecting the referents of
> the model to entities in the world and for recognizing when they do and
> don't apply. Moreover, on nearly all problems to which more than one of
> them is applicable, when applied by a competent modeler, they give nearly
> indistinguishable answers to most questions of practical modeling interest.
>
> I have heard exactly one proposed ontology for fuzzy membership functions
> (proposed by Judea Pearl, among others) that makes sense to me. Under this
> proposed ontology, the fuzzy membership of Robbie's adult height in the set
> "tall," in a given context, should be taken as proportional to the
> probability that a generic person in that context would use the label
> "tall" to describe Robbie. Thus, fuzzy memberships are likelihood
> functions. We can think of them as soft evidence applied to numerical
> crisp set height measurements.
>
> I might go beyond this and suggest an alternate criterion, that the fuzzy
> membership be proportional to the *utility* for an appropriately defined
> decision maker in that context of using the term "tall" to describe Robbie
> (this, for example, would allow us to weigh costs of inappropriate usage of
> the term).
>
> This proposed ontology makes a lot of sense at a surface level. However, I
> know it is not what most fuzzy set researchers think they are talking about
> when they use fuzzy memberships. I've never seen its mathematical
> implications worked out, or seen any discussions about whether or under
> what circumstances it gives rise to combination rules that look anything
> like what the fuzzy people now use.
>
> I therefore find myself in the difficult position of being highly
> sympathetic to the concerns that drove people to invent fuzzy sets in the
> first place, but extremely skeptical about whether what they've developed
> solves the problem they set out to solve in an acceptable way.
>
> Kathy Laskey
- --
Taner Bilgic taner@boun.edu.tr
Department of Industrial Engineering Tel: +90-212-263 1540 x2078
Bogazici University Fax: +90-212-265 1800
Bebek TR-80815 Istanbul Turkey http://www.ie.boun.edu.tr/~taner
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