**Subject: **RE: Probability and possibility

**From: **Makropoulos, Christos (*c.makropoulos@ic.ac.uk*)

**Date: **Thu Nov 09 2000 - 18:28:11 MET

**sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Larry Hall: "IFSA/NAFIPS 2001 Final call for papers"**Previous message:**Stephan Lehmke: "CfP: 7th Fuzzy Days, Dortmund, Germany, Oct 1-3, 01"**Maybe in reply to:**Jean-Philippe Drecourt: "Probability and possibility"**Next in thread:**Gert de Cooman: "Re: Probability and possibility"**Maybe reply:**Makropoulos, Christos: "RE: Probability and possibility"

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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**Next message:**Larry Hall: "IFSA/NAFIPS 2001 Final call for papers"**Previous message:**Stephan Lehmke: "CfP: 7th Fuzzy Days, Dortmund, Germany, Oct 1-3, 01"**Maybe in reply to:**Jean-Philippe Drecourt: "Probability and possibility"**Next in thread:**Gert de Cooman: "Re: Probability and possibility"**Maybe reply:**Makropoulos, Christos: "RE: Probability and possibility"

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