RE: Probability and possibility


Subject: RE: Probability and possibility
From: Marco A. Vera (m-vera@uniandes.edu.co)
Date: Wed Nov 15 2000 - 21:24:59 MET


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Hi Jean-Philippe:

Attached please find some mails posted on the list related to your
question. I'd like to emphasize that Fuzzy sets are an approximation
to
possibility modeling, so the discussion is better about Fuzzy Theory
vs. Probability Theory.

Best regards,

Marco A. Vera
University of los Andes
Bogotá - Colombia

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From: Gert de Cooman <gert.decooman@rug.ac.be>
Reply-To: gert.decooman@rug.ac.be
To: Multiple recipients of list <fuzzy-mail@dbai.tuwien.ac.at>
Subject: Re: fuzzy and probability
Date: Thu, 4 Feb 1999 18:54:00 +0100 (MET)
=20

> Can anyone recommend a book, (or an article) that has a
> good discussion on the difference between fuzzy set theory
> and probability theory.
The following papers may be of interest: they are about the connection
between possibility theory (and fuzzy set theory) and probability theory
in the context of the theory of imprecise probabilities. You can find
more information on my website (http://ensmain.rug.ac.be/~gert) under
the heading "publications".
More information about imprecise probabilities can be found on the
website of the Imprecise Probabilities =
Project(http://ensmain.rug.ac.be/~ipp).
---------------------
Peter Walley and Gert de Cooman, A behavioural model for linguistic
uncertainty, 26 pages, accepted for publication in: Computing with
Words, ed. Paul P. Wang, 1998.=20
Gert De Cooman, Integration in possibility theory, 40 pages, submitted
for publication in: Fuzzy Measures and Integrals - Theory and
Applications, ed. M. Grabisch, T. Murofushi en M. Sugeno, 1998.=20
Peter Walley and Gert de Cooman, Coherence of rules for defining
conditional possibilities, 29 pages, accepted for publication in
International Journal of Approximate Reasoning, 1998.=20
Gert de Cooman and Peter Walley, An imprecise hierarchical model for
behaviour under uncertainty, 34 pages, submitted for publication to
Econometrica, 1998.=20
Gert de Cooman and Dirk Aeyels, A random set description of a
possibility measure and its natural extension, 6 pages, submitted for
publication in IEEE Transactions on Systems, Man and Cybernetics, 1997.=20
Gert de Cooman and Dirk Aeyels, Supremum preserving upper probabilities,
27 pages, accepted for publication in Information Sciences, 1998.=20
-------------------Best wishes,Gert de Cooman--=20
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
Dr. ir. Gert de CoomanPostdoctoraal Onderzoeker FWO/Postdoctoral Fellow =
FWO
--------------------------------------------------------------
E-mail: gert.decooman@rug.ac.beURL: http://ensmain.rug.ac.be/~gert
--------------------------------------------------------------Universitei=
t Gent
Vakgroep Elektrische EnergietechniekOnderzoeksgroep SYSTeMS
Technologiepark - Zwijnaarde 99052 Zwijnaarde Belgium
--------------------------------------------------------------
Tel: +32-(0)9-264 56 53Fax: +32-(0)9-264 58 40
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
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Re: Transfoming probability distributions into fuzzy sets - can anyone =
help?
J. Lawry (enjl@PROBLEM_WITH_YOUR_MAIL_GATEWAY_FILE)
Fri, 11 Sep 1998 02:03:45 +0200 (MET DST)=20
    Messages sorted by: [ date ][ thread ][ subject ][ author ]=20
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    classification"=20
Carlos Gershenson (carlos@jlagunez.iquimica.unam.mx) wrote:
: On Mon, 17 Aug 1998, Anthony Cowden wrote:
:=20
: > WSiler wrote:
: > >=20
: > > >While I agree that we should not rule out a relationship between =
fuzzy
: > > >sets and probability ( indeed I am a strong advocate of =
probabilistic
: > > >semantics for fuzzy sets) I do not agree that we should take =
probability
: > > >distributions of random variables (normalised or not) as =
membership
: > > >functions of fuzzy sets. The former quantify uncertainty =
regarding the
: > > >value of a random variable and the other vagueness of definition.
: > > >
: > > It is certainly true that "probability distributions quantify =
uncertainty
: > > regarding the value of a random variable", to say that =
"[membership=20
functions
: > > of fuzzy sets characterize] vagueness of definition" is a quite=20
unnecessary
: > > restriction on fuzzy sets. Having worked on real-world =
applications of=20
fuzzy
: > > expert systems for some fifteen years now, I consider that fuzzy =
sets can
: > > characterize uncertainty of whatever origin, including both =
vagueness and
: > > values of random variables among many others.
: > >=20
: > > To assert that a normal distribution characterizes a numeric =
random=20
variable
: > > subject to a large number of small errors amounts to a tautology,=20
parameterized
: > > perhaps as a mean and variance. However, I can (and often do) =
characterize=20
that
: > > same variable as a bell-shaped fuzzy number, paramaterized perhaps =
as=20
central
: > > value and a hedge "roughly". There is no vagueness here, just an=20
uncertainty as
: > > to precise value. In an expert system, "roughly 2" is a heck of a =
lot more
: > > useful than "2 +/- 25%".
: > >=20
: > > A list of the kinds of uncertainty which can be fruitfully =
represented by=20
fuzzy
: > > quantities (e.g. truth values of scalars, fuzzy numbers, =
membership=20
functions,
: > > truth values of rules, truth values of members of a discrete fuzzy =

set,...)
: > > would probably be quite long. If I'm not sure that a car is a Ford =
or a
: > > Chevrolet, that uncertainty is easily represented by the grades of =

membership
: > > in a discrete fuzzy set of car makes, for example.
: >=20
: > Bill:
: >=20
: > Thanks for the automobile lead-in...
: >=20
: > To help me understand some of the points raised, allow me to pose a
: > problem:
: >=20
: > I own a Mercury Villager mini-van, which is made in the same factory =
as
: > the Nissan Quest (in Ohio, by the way), and most of the parts are
: > identical and interchangeable. As you might assume, they look very
: > similar. Now, if I see 2 mini-vans in a parking lot, and they appear =
to
: > be a Villager/Quest, but I can't tell from the distance I am at, =
than
: > the probability that the one on the left is a Villager is .5, and =
the
: > probability that it is a Quest is .5 (the same goes for the one on =
the
: > right).
: >=20
: > Now, if I walk out into the parking lot and inspect the 2 vehicles, =
I
: > find that the one on the left is a Quest and the one on the right is
: > also a Quest. The probability now is 0.0 that either one is a
: > Villager. But what about the membership in the set (classification,
: > identity, whatever) of Villager? I would say that the Quest has a
: > membership of .95 in the set of Villager (and vice versa). How does
: > probability help explain to a mechanic that he can fix a Villager if =
he
: > has only ever fixed Quests before?
: >=20
: > Tony
:=20
: In the problem you propose, you would need to use "similarity". The =
more
: similarity there is between 2 elements, the less they exculde each =
other.
:=20
: This is why a mechanic can fix a Villager the first time he sees one.
: Because it is very similar to the Quest, which he is used to.
:=20
:=20
:=20
: >=20
: > >=20
: > > I'm not sure what latitude FRIL offers in the kinds of things =
which can be
: > > represented by fuzzy quantities, but I surely hope it covers more =
than=20
vague
: > > definitions.
: > >=20
: > > William Siler
: > >=20
: >=20
: > --=20
: > =
*********************************************************************
: > Anthony Cowden, Manager, Fuzzy Systems Solutions=20
: > Sonalysts, Inc.
: > Fuzzy Systems Solutions: http://www.sonalysts.com/fuzzy.html
: > Fuzzy Query (TM): http://www.sonalysts.com/fq.html
: >=20
: >=20
: > =
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: > This message was posted through the fuzzy mailing list.
: > (1) To subscribe to this mailing list, send a message body of
: > "SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at
: > (2) To unsubscribe from this mailing list, send a message body of
: > "UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL =
yoursubscription@email.address.com"=20
: > to listproc@dbai.tuwien.ac.at
: > (3) To reach the human who maintains the list, send mail to
: > fuzzy-owner@dbai.tuwien.ac.at
: > (4) WWW access and other information on Fuzzy Sets and Logic see
: > http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info
: > (5) WWW archive:=20
http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html
: >=20
:=20
: "There is no Truth but that of Eternal struggle..."
: -Orunlu the Keeper
:=20
: Carlos Gershenson
: http://132.248.11.4/~carlos/
:=20
:=20
: =
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: (2) To unsubscribe from this mailing list, send a message body of
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: to listproc@dbai.tuwien.ac.at
: (3) To reach the human who maintains the list, send mail to
: fuzzy-owner@dbai.tuwien.ac.at
: (4) WWW access and other information on Fuzzy Sets and Logic see
: http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info
: (5) WWW archive: =
http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html
:=20

I'm not sure that this is that good an example of a problem that cannot =
be
modelled by probability theory. We could after all consider the
probability that a component picked at random from a Villager was
identical to the same component (i.e. component with the same
function) in the Quest. If Villager and Quest are indeed 'similar' then
this could be 0.95.

Jonathan Lawry=20

--=20
Dr Jonathan Lawry,
AI Group,
Dept. Engineering Mathematics,
University of Bristol,
Queens Building,
University Walk,
Bristol, BS8 1TR, UK

Email:j.lawry@bristol.ac.uk
Tel:+44 117 928 8184

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Next message: Rob North: "On the wrong horse?"=20
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Re: Transfoming probability distributions into fuzzy sets - can anyone =
help?
Fred A Watkins (fwatkins@hyperlogic.NO_SPAM.com)
Mon, 17 Aug 1998 20:56:41 +0200 (MET DST)=20
    Messages sorted by: [ date ][ thread ][ subject ][ author ]=20
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book"=20
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wsiler@aol.com (WSiler) wrote:

>>Just a general comment that I think it is best to keep fuzzy logic and
>>probability theory as far removed as possible!=20
>
>I realize than many or most fuzzy persons share your viewpoint here, =
but I do
>not. Both probability distributions and membership functions look from
>different vantage points ar pretty much the same thing - uncertain =
numbers,
>whatever the cause of the uncertainty may be. A very few persons have =
worked
>seriously on the relationship between the two, which I think is =
unfortunate.

There are mathematical relations (see below) because both fields grind =
over
the same territory: R, the reals (usually, at least, because the input =
to the
system is commonly a *measurement*) and I, the unit interval. But the =
key point
of difference is philosophical. Fuzziness is ambiguity occasioned by the =
use
of words; since words want *definition*, and definitions are impossible =
to
obtain in the realm of experience, to use a word to refer to external =
reality
is to be imprecise. The degree of imprecision is measured as degrees of=20
fuzziness.
On the other hand, probability describes lack of knowledge of *well =
defined*
events. To summarize: fuzziness is lack of information in *meanings*,=20
probability
is lack of information in *occurrences". Of course, "well defined =
events" are,
strictly speaking, unavailable because of the difficulty in applying =
definitions
to events; simplification is always required. Finally, if one wants to=20
investigate
probability as a description of *belief*, OK, but now the treatment is a =

mathematical
abstraction (admitting "well-defined events") that has unclear relation =
to=20
reality.
A famous remark by Einstein captures this idea very well.

>There is, in any event, a quick (if incomplete and inaccurate) answer =
to the
>question. Probability density functions have area one; fuzzy numbers =
and
>membership functions usually have max value one. Simply normalize the
>probability distribution to a max value of one, and there is your fuzzy =
number
>or membership function. If the probability distribution is discrete, do =
the
>same thing, and there is your discrete fuzzy set.

Let's tighten up just a bit. The typical definition for a *distribution=20
function"
over the real line is

F(x) =3D P((-infinity, x])

where P is the underlying probability measure. Since F is defined in =
terms of P,
it inherits its min and max from P (i.e., 0 and 1). Since the measures =
of sets
in a nested sequence of measurable sets are monotone, so is F, since for
any x < y we have (-infinity, x] is included in (-infinity, y].

The above implies that any distribution *function* is in fact a fuzzy =
set,=20
because
a fuzzy set is simply a map from the "universe" (here it's R) into I =3D =
[0, 1],
without other conditions.

But distribution functions are not particularly valuable linguistically. =
For=20
example,
while a Gaussian about zero might describe "(approximately) zero", the=20
distribution function
that density generates (the "erf" function) has no meaningful relation =
to the=20
word "zero".

What most people want to do is start with probability *densities*. A =
density f=20
(over R)
is a non-negative integrable function such that Integral_over_R(f) =3D =
1. Given a=20
density
f, one recovers its distribution F by integration:

F(x) =3D Integral_from_-infinity_to_x(f(t) dt)

It's not always possible to obtain a density from a distribution because =
the=20
derivative
of the distribution may not exist. Further assumptions can be made to =
help with=20
this.

Anyway, the idea now is to relate densities to fuzzy sets. As previously =

remarked, a
density is integrable, but integrability does not imply boundedness. =
This means=20
that
the suggested normalization is not generally applicable. But when a =
density has=20
a
finite supremum (least upper bound) this normalization is possible and =
the=20
normalized
density is a fuzzy set *by definition*. On the other hand, a fuzzy set =
need not=20
be
integrable (remember "erf"?) and so a fuzzy set cannot in general be =
converted=20
into
a probabilty density without loss of data (say, by truncation). =
Moreover, a=20
fuzzy set
need not be monotone and so is not in general a distribution function.

To summarize: a fuzzy set derives a probability density precisely when =
that set=20
is
integrable; a probability density derives a fuzzy set precisely when it =
is=20
bounded.

>The question of the circumstances under which that simple answer is =
wrong is
>interesting, and comments on this would be welcome. Am I wide open?

It's really pretty straightforward. Hope this helps.

Fred A Watkins, Ph.D.
HyperLogic Corporation
PO Box 300010
Escondido, CA 92030-0010 USA
voice: +1 760 746 2765 x 9117
fax: +1 760 746 4089
email: fwatkins@hyperlogic.SNIP_THIS_OUT.com

Decode email address to contact me.

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(4) WWW access and other information on Fuzzy Sets and Logic see
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(5) WWW archive: =
http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html

    Next message: Michael Rolfe: "Software problems, McNeill&Thro's =
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From: Ellen Hisdal <ellen@ifi.uio.no>
Reply-To: ellen@ifi.uio.no
To: Multiple recipients of list <fuzzy-mail@dbai.tuwien.ac.at>
Subject: WHAT IS FUZZY SET THEORY? Re: fuzzy and probability
Date: Wed, 24 Feb 1999 22:38:04 +0100 (MET)
                       =20
                           =20
Response to email letter to fuzzy group from Nan-Chieh Chiu
                                             njchiu@eos.ncsu.edu
                                             Operations Research
                                             North Carolina State =
University
                                             Sat, 30 Jan 1999
Subject:Fuzzy and probability.=20
        Also Lotfi's email letter to the fuzzy group,
        and the discussion following it at the end of 1998.
        WHAT IS FUZZY SET THEORY?
> Can anyone recommend a book, (or an article) that has a
> good discussion on the difference between fuzzy set theory
> and probability theory.=20

Dear Nan-Chieh Chiu,

Your question is a very important one. It has to do with the existence
of a serious ambiguity as to what fuzzy set theory is. This ambiguity
was also illustrated by Lotfi Zadeh's letter to the fuzzy group,=20
and the discussion following it at the end of 1998.

Personally I consider fuzzy set theory and fuzzy logic to be a theory
which tries to explain how human beings operate with linguistic values
of variables such as `tall', `young', `very young' etc..=20
And how human beings can assign partial grades of membership=20
(mu element of the real interval [0,1]) to an object=20
or to a numerical attribute value u in a class. For example,=20
a grade of membership of 0.5 to a height u=3D170cm in the class `tall =
woman'.

I think that all Fuzzy Settians will agree with me that the above
subject is an important part of Fuzzy Set theory; Lotfi Zadeh deserves
much honor for introducing the concept of a partial grade of membership
of an object (or attribute value) in a class in order to explain how
humans think and reason with the aid of linguistic values of variables.=20
Furthermore he deserves honor for drawing our attention to the
importance of taking uncertainty into account in all cases (everyday
and technological ones) in which exact values are unavailable.

However, for many Fuzzy Settians, there exists another part of=20
fuzzy set theory. This is the claim that the concept of a partial
grade of membership of an object or an attribute value in a class
does not have a probabilistic interpretation. I consider this claim
to be a pure DOGMA which is not worthy of a serious scientific theory.
The reason for this is that a careful probabilistic interpretation=20
of grades of membership gives more generally reasonable results for =
labels=20
with connectives (such as the label `tall OR medium') than the max-min
operators which LZ (Lotfi Zadeh) suggests for the connectives
(such as the max operator for OR, and min for AND).

Somewhat superficially stated, LZ's max-min fuzzy set theory replaces
the + and x (times) operators of probability theory by max and min
respectively. These operations give reasonable results in some cases,
such as for `tall OR VERY tall', and unreasonable ones in others
(e.g. for `tall OR medium' (OR=3Dinclusive OR)=20
whose `max' membership curve has a sharp dip around the boundary between =

`tall' and `medium').

The unreasonable cases disappear in the TEE model for grades of =
membership
(see ref [1] below). Again very superficially stated, the TEE model =
interprets=20
mu (u=3D170cm) tall (mu with subscript `tall') as P(tall|u), the =
probability=20
that a woman of height 170cm will be assigned the label tall by a =
subject.

This is in contrast to the usual comparison performed by Fuzzy Settians
between the grade of membership for `tall' on the one hand,=20
and P(u|tall) on the other. P(u|tall woman) is the probability that a
woman labeled `tall' has a measured numerical height value u.

It IS true that P(u|tall) and the grade of membership of u in the
class `tall (woman)' have generally quite different numerical values.=20
So do P(u|tall) and P(tall|u) for the same height u.=20

(P(u|tall) must sum up to 1 over all intervals in a (quantized) U =
universe
(or integrate to 1 in a continuous U universe). This does not hold for
P(tall|u)=3D[mu(u) for `tall']. P(tall|u) must, for a given u, sum up to =
1
over all height labels which a subject uses.)

Because of the unreasonable values which can occur in LZ's max-min =
theory
for the connectives, many other operators for the connectives have
been have been suggested instead of max and min in the course of time.
Except for the TEE model, all the other suggestions carefully avoid a
probabilistic interpretation because of THE DOGMA. This sad dogmatic
attitude should be unacceptable from a scientific point of view.

I hope that this summary of the situation may be of some help to you.

Let me mention that some researchers deal with the extra complication
of replacing numerical probability values by linguistic ones. For =
example,
they may replace an approximate probability value of 0.95=20
by the linguistic value `very big', and use a membership curve for `VERY =
big'.=20
However, this complicated question has no connection with the basic =
issues
mentioned in this letter.

Finally let me add that fuzzy set theory and fuzzy logic also try to
be a generalization of classical logic (propositional calculus).
Instead of operating only with truth values True (grade of
membership 1) and False (grade of membership 1), truth values between 0
and 1 are also accepted in fuzzy logic. Such intermediate truth values
have been used before in many valued-logics. The best known of these
is perhaps the one due to Lukasiewicz [2] (pages 87 and 131),=20
It has quite a bit of similarities to fuzzy logic.

A logic which completely carries through the analogy between truth
values and probabilities is described in reference [3].

Best greetings,
                  Ellen Hisdal

---------------------------------------------------------------------
Address, etc.: =20
       Ellen Hisdal | Email: ellen@ifi.uio.no
       (Professor Emeritus) |
Mail: Department of Informatics | Fax: +47 22 85 24 01
       University of Oslo | Tel: (office): 47 22 85 24 39
       Box 1080 Blindern |
       0316 Oslo, Norway | Tel: (secr.): 47 22 85 24 10
Location: Gaustadalleen 23, | =20
       Oslo | Tel: (home): 47 22 49 56 53
---------------------------------------------------------------------

                    REFERENCES

[1]
@incollection{ruan,
   author =3D {Hisdal, E.},
   title =3D {Open-Mindedness and Probabilities versus Possibilities},
   booktitle=3D{Fuzzy Logic Foundations and Industrial Applications},
   publisher =3D {Kluwer Academic Publishers, Boston},
   year =3D {1996},
   editor =3D {Da Ruan},
   pages =3D {27-55} }

[2]
@book{borkowsky,
   editor =3D {Borkowsky, L.},
   title =3D {Jan Lukasiewicz, Selected Works},
   publisher =3D {North Holland},
   year =3D {1970} }

[3]
@book{hisdal,
   author =3D {Hisdal, Ellen},
   title =3D {Logical Structures for Representation of Knowledge=20
              and Uncertainty},
   publisher =3D {Physica Verlag, A Springer-Verlag Company},
   year =3D {1998} }
             =20

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From: Will Dwinnell <76743.1740@compuserve.com> Save Address Block =
Sender
Reply-To: 76743.1740@compuserve.com
To: Multiple recipients of list <fuzzy-mail@dbai.tuwien.ac.at>
Subject: Re: WHAT IS FUZZY SET THEORY? Re: fuzzy and probability
Date: Wed, 3 Mar 1999 01:23:55 +0100 (MET)

"However, for many Fuzzy Settians, there exists another part of=20
fuzzy set theory. This is the claim that the concept of a
partial grade of membership of an object or an attribute value in=20
a class does not have a probabilistic interpretation. I consider=20
this claim to be a pure DOGMA which is not worthy of a serious=20
scientific theory."

I can only speak for myself ('fuzzy settians' with other ideas=20
are on their own), but to my way of thinking, fuzzy logic is less=20
a 'scientific theory' than it is a useful engineering tool. =20
Comparisons to deductive logic and probability theory are=20
important, but I do not think they define fuzzy logic well. I=20
prefer to consider what role fuzzy logic plays, and how it does=20
so to understand what it is. At the bottom level, fuzzy logic=20
uses mathematical distributions (yes, like probability=20
distributions). Whether we wish to give the numbers which=20
comprise those distributions a probabilistic interpretation would=20
seem to be a matter of context, not 'dogma', to me.

--=20
Will Dwinnell

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From: njchiu@unity.ncsu.edu (Nanchieh Jay Chiu) Save Address Block =
Sender
Reply-To: njchiu@unity.ncsu.edu
To: Multiple recipients of list <fuzzy-mail@dbai.tuwien.ac.at>
Subject: Re: fuzzy and probability
Date: Wed, 24 Feb 1999 20:22:50 +0100 (MET)
                       =20
            =20
>Can anyone recommend a book, (or an article) that has a
>good discussion on the difference between fuzzy set theory
>and probability theory.=20
>

Hello:
=20
This is a follow-up message to the "fuzzy and probability"=20
question posted on 1/27. I'd like to thank all the people=20
who responded to the question. A list of the results is=20
summarized here. They are (in the order I received):

-- from Lofti Zadeh
 ...check out Bart Kosko's work - or potentially his progenitor.

-- from Juite (Ray) Wang
Laviolette, M. M., Seaman,J.W., Jr., Barrett, J.D., and Woodall, W.H.
 (1995) "A Probabilistic and Statistical View of Fuzzy Methods."=20
Technometrics, Vol. 37, No. 3, pp. 249-292.

Special issue papers on the effect of fuzzy representations, IEEE
Transactions on Fuzzy Systems, Vol. 2, No. 1.

-- from Olivier H=E9lary
 See D.DUBOIS on www.irit.fr

-- from Maurice Girod
 'Fuzziness and Probability' by S.F.Thomas
 ISBN 0-9649049-0-X

-- from Gert de Cooman
 The following papers may be of interest: they are about the connection
 between possibility theory (and fuzzy set theory) and probability =
theory
 in the context of the theory of imprecise probabilities. You can find
 more information on my website (http://ensmain.rug.ac.be/~gert) under
 the heading "publications".

 More information about imprecise probabilities can be found on the
 website of the Imprecise Probabilities Project
 (http://ensmain.rug.ac.be/~ipp).
---------------------
 Gert de Cooman and Peter Walley, A behavioural model for linguistic
 uncertainty, 26 pages, accepted for publication in: Computing with
 Words, ed. Paul P. Wang, 1998.=20

 Gert De Cooman, Integration in possibility theory, 40 pages, submitted
 for publication in: Fuzzy Measures and Integrals - Theory and
 Applications, ed. M. Grabisch, T. Murofushi en M. Sugeno, 1998.=20

 Gert de Cooman and Peter Walley, Coherence of rules for defining
 conditional possibilities, 29 pages, accepted for publication in
 International Journal of Approximate Reasoning, 1998.=20

 Gert de Cooman and Peter Walley, An imprecise hierarchical model for
 behaviour under uncertainty, 34 pages, submitted for publication to
 Econometrica, 1998.=20

 Gert de Cooman and Dirk Aeyels, A random set description of a
 possibility measure and its natural extension, 6 pages, submitted for
 publication in IEEE Transactions on Systems, Man and Cybernetics, 1997. =

 Gert de Cooman and Dirk Aeyels, Supremum preserving upper =
probabilities,
 27 pages, accepted for publication in Information Sciences, 1998.=20

-- from Henri Prade
   (papers written by Henri Prade and Didier Dubois)=20

Thank you.

Nan-Chieh Chiu
njchiu@eos.ncsu.edu
Operations Research
North Carolina State University
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