# RE: Probability and possibility

Subject: RE: Probability and possibility
From: Makropoulos, Christos (c.makropoulos@ic.ac.uk)
Date: Thu Nov 09 2000 - 18:28:11 MET

dear jean-philippe,

your question is always a fundamental question to keep in mind:
this answer is not my own and was based on:
http://www.cs.cmu.edu/Web/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html, where
you might want to look for more general information.

The question has to be answered in two ways: first, how does fuzzy
theory differ from probability theory mathematically, and second, how
does it differ in interpretation and application.

At the mathematical level, fuzzy values are commonly misunderstood to be
probabilities, or fuzzy logic is interpreted as some new way of handling
probabilities. But this is not the case. A minimum requirement of
probabilities is ADDITIVITY, that is that they must add together to one, or
the integral of their density curves must be one.

But this does not hold in general with membership grades. And while
membership grades can be determined with probability densities in mind (see
[11]), there are other methods as well which have nothing to do with
frequencies or probabilities.

Because of this, fuzzy researchers have gone to great pains to distance
themselves from probability. But in so doing, many of them have lost track
of another point, which is that the converse DOES hold: all probability
distributions are fuzzy sets! As fuzzy sets and logic generalize Boolean
sets and logic, they also generalize probability.

In fact, from a mathematical perspective, fuzzy sets and probability exist
as parts of a greater Generalized Information Theory which includes many
formalisms for representing uncertainty (including random sets,
Demster-Shafer evidence theory, probability intervals, possibility theory,
general fuzzy measures, interval analysis, etc.). Furthermore, one can
also talk about random fuzzy events and fuzzy random events. This whole
issue is beyond the scope of this FAQ, so please refer to the following
articles, or the textbook by Klir and Folger (see [16]).

Semantically, the distinction between fuzzy logic and probability theory
has to do with the difference between the notions of probability and a
degree of membership. Probability statements are about the likelihoods of
outcomes: an event either occurs or does not, and you can bet on it. But
with fuzziness, one cannot say unequivocally whether an event occured or
not, and instead you are trying to model the EXTENT to which an event
occured.

The classic example: someone is or in not 1.80m of height. You can associate
a probability of finding someone 1.80m in a group of people. But is he
"tall"? You cant associate a probability because there is abiguity as to the
meaning of outcome itself (more importantly there is some ambiguity as to
the rules deciding when someone is or is not tall, or better still when he
somewhere is between. The applications of this are very interesting:
consider the meaning of the words vulnerability, suitability, etc. (to take
some examples from the environmental and civil engineering domain)

hope this helps
best regards

Christos

__________________________________________________
christos k. makropoulos

environmental & water resources engineering
research group

civil engineering department
imperial college of science, technology & medicine
london SW7 2AZ
united kingdom
office: ++207-5946018
fax: ++207-2252716

-----Original Message-----
From: Jean-Philippe Drecourt [mailto:jpd@dhi.dk]
Sent: Wednesday, November 08, 2000 2:03 PM
To: Multiple recipients of list
Subject: Probability and possibility

Dear group members,

I would like to get a clear explanation of the difference between possiblity
measure and probability measure. In other words, I would like to know the
answer to the question: "What possibility can do that probability cannot".

This is not an attempt to argue against fuzzy logic. I have just never found
a good explanation.

Thanx

Jean-Philippe

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