Re: Fuzzy logic compared to probability

Mark A. Scarton (marksc@wpmail.code3.com)
Tue, 5 Mar 1996 17:40:24 +0100


poncet@isi.ee.ethz.ch (Andreas Poncet) wrote:
>Well, usually, in the papers about fuzzy logic, it is claimed that fuzzy
>membership functions are not to be mixed up with probabilities.
>But what kind of probabilities? The arguments used are typically that
>a "membershipness" can be given for any single observation, whereas no
>"probability" - in the sense of frequency - can be defined. Fine, but
>what if we consider the Bayesian subjective view of probability, namely
>as a degree of belief in (or a degree of relevance of) a given
>proposition? Then it seems to me that a "membershipness" is in fact
>a CONDITIONAL PROBABILITY.

In my work in the application of expert technology in medicine, one of the
principle aspects of the difference between probability and membership is that
the former is based upon an closed world model, requires independence, and
assumes additivity and monotonicity, while the latter is based upon an open
world model and makes far fewer assumptions. The data that we deal with is much
less "pure" and too much is unknown (about interactions, for example).

To relate a point discussed just yesterday between an collegue and myself
regarding an expert system for medical informatics. If facts are being
asserted within a fuzzy frame of reference, the absence of a fact does not imply
knowledge. In a probabilistic framework, it would since in a system based upon
a closed world model it is assumed that everything is known.

The assumptions underlying membership are much closer to the basis of "fact"
being asserted by a physician than a probability. My rule of thumb is that when
the basis for a probabilistic statement of fact can be made, use a measure of
probability. I.e. if you can live within the constraints of the probabilistic
system, use its framework. Just don't do so by abstracting the real world
entity being modelled so far from reality that the assertion no longer
represents the real world. Again, my experience as an engineer has been that
the more complex the system is that is being modelled or the more that it
includes human action or thought, the more dangerous it is to constrain the
system to the assertion of probabilistic evidence.

I must agree with what I see to be an underlying thread of argument by the
original poster and the follow-up. IMHO the origin of a particular membership
value in fuzzy theory is much less understood than the origin of a value in a
probabilistic system. We need methodologies for the automated derivation and/or
assignment of membership value. Obviously, if you are working strictly from
evidence, a statistical or probabilistic measure lends itself. So how would
that value be "fuzzified"?

Please don't misconstrue my last paragraph to indicate that such methods don't
exist now. I am simply ignorant of them. Maybe someone else can help out with
references?
Mark A. Scarton, ABD
CompUtah!, Park City, Utah USA
Home: 801.565.9835
Office: 801.265-4612