Re: Probability-Possibility Consistency

Ellen Hisdal (ellen@ifi.uio.no)
Fri, 28 Nov 1997 20:23:51 +0100 (MET)

On Sun, 9 November 97, Franck DELCROIX wrote:

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> Hi !
>
> Just a little question about the so-called "consistency principle"
>
> If I'm right, it's a set of rules describing a relation between
> probability and possibility.
>
> But how does it works in a practical sense ?
>
> I mean, if I have a given probability distribution, how can I express
> the "corresponding" possibility distribution ?
>
>
> Can some fuzzy gurus help me by giving some good references (books,
> papers, URL, ...) or exemples ?
>
> Thanks a lot,
>
> Franck.
>
Dear Franck,

In his 1978 possibility paper (see reference [7])
Zadeh defines the consistency
of a probability distribution Prob(ui)
with a possibility distribution Poss(ui)
as the sum of the products Prob(ui)Poss(ui) over all attribute values ui
(e.g. height values u quantized in cm intervals). This quantity lies
always between 0 and 1 because Prob(ui) sums up to 1.

Let `a' be the label of a fuzzy set, e.g. `tall' or `short'.
Both Prob(ui) and Poss(ui) must refer to this label. The TEE model for
grades of membership interprets these as follows:

Prob(ui)=P(ui|a)=probability that an object which has been labeled yes-a
by a subject has the (exactly measured) height ui.

Poss(ui)=Prob(a|ui)=probability that a subject will assign to an object of
(exactly measured) height ui the label yes-a.

The reason why Poss(ui)=Prob(a|ui)
is not always 1 or 0 (which it would be if the
subject had used exact, nonfuzzy threshold for `a' (tall)) (in the
universe U of exactly measured height values) is the
presence of one or more sources of fuzziness (see [1]). One of these
sources can be that a subject (who assigns the grade of membership
or possibility value in the fuzzy set `a=tall' to an object
whose exact height ui is specified to him) takes into account
that other subjects only estimate the height value of an object in
everyday life. An object of exact height ui may thus be estimated by
another subject to have a slightly different height.
Another source of fuzziness can be that the
subject (who assigns the membership or possibility value
in a=tall to the object of exact height ui) takes into account
that other subjects have slightly variable thresholds in U (universe
of exactly measured height values)
for the exact height value of an object to which they would assign the label
yes-a. (A third source of fuzziness is mentioned in [1]).

It turns out that Zadehs `possibility-probability consistency'
is equal to P(a|a),
the probability that an object labeled `a' in one experiment will be
labeled `a' again in a second experiment (assuming that the conditions
of observation of the object are chosen at random from a set of
conditions of observation in each of the two experiments. Or that the
assumed subject who assigns the label `a' is chosen at random from a
set of subjects with somewhat different thresholds for, e.g. a=tall).
This is proved in eq.(12.12), page 12.3, of reference [2]
and discussed in eq.(12.20) of the same paper in connection with the
possibility-probability consistency.

These subjects are discussed in a concentrated form in
[8, page 108, eq.(21)], [3] and [4],
and in more detail in [4] and [5] and [2]. You can obtain the research
reports by writing an ordinary letter to the Institute of Informatics.

[1]
@article{inf1p2,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.2. {D}ifferent Sources of Fuzziness
and Uncertainty},
journal = {Int. J. Man-Machine Studies},
year = {1986},
volume = {25},
pages = {113-138} }

[2]
@techreport{64,
author = {Hisdal, E.},
title = {A Theory of Logic Based on Probability},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1984},
type = {Research Report},
note = {ISBN~82-90230-60-5},
number = {64} }

[3]
@article{are,
author = {Hisdal, E.},
title = {Are Grades of Membership Probabilities?},
journal = {Fuzzy Sets and Systems},
year = {1988},
volume = {25},
pages = {325-348} }

[4]
@incollection{ruan,
author = {Hisdal, E.},
title = {Open-Mindedness and Probabilities versus Possibilities},
booktitle={Fuzzy Logic Foundations and Industrial Applications},
publisher = {Kluwer Academic Publishers, Boston},
year = {1996},
editor = {Da Ruan},
pages = {27-55}
}

[5]
@techreport{inf1p3,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.3. {R}eference Experiments and
Label Sets},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-053-3.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-3.ps},
number = {147} }

[6]
@techreport{inf1p4,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.4. {T}he {TEE} Model},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-054-1.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-4.ps},
number = {148} }

[7]
@article{zposs,
author = {Zadeh, L.A.},
title = {Fuzzy Sets as a Basis for a Theory of Possibility},
journal = {Fuzzy Sets and Systems},
year = {1978},
volume = {1},
pages = {3-28} }

[8]
@article{inf1p1,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.1. {D}ifficulties with Present-Day
Fuzzy Set Theory and their Resolution in the {TEE} Model},
journal = {Int. J. Man-Machine Studies},
year = {1986},
volume = {25},
pages = {89-111},
ignored = {page 94 for different words for grade of membership
page 95 for lack of difference between distr tall|u and u|tall},
}

Best greetings
Ellen Hisdal

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