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
This archive was generated by hypermail 2b29 : Mon Feb 28 2000 - 06:38:30 PST