Re: Probability-Possibility Consistency
(enjl@zeus.bris.ac.uk)
Sun, 7 Dec 1997 22:39:25 +0100 (MET)
Ellen Hisdal (ellen@ifi.uio.no) wrote:
:
: On Sun, 9 November 97, Franck DELCROIX wrote:
:
: > Reply-To: franck.delcroix@devinci.fr
: > Originator: fuzzy-mail@dbai.tuwien.ac.at
: > Sender: fuzzy-mail@dbai.tuwien.ac.at
: > Precedence: bulk
: > From: Franck DELCROIX <franck.delcroix@devinci.fr>
: > X-Listprocessor-Version: 6.0c -- ListProcessor by Anastasios Kotsikonas
: > X-Comment: Fuzzy Distribution List
: > Content-Transfer-Encoding: 7bit
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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
:
:
: ---------------------------------------------------------------------
: Address, etc.:
: Ellen Hisdal | Email: ellen@ifi.uio.no
: (Professor Emeritus) |
: Mail: Department of Informatics | Fax: +47 22 85 24 01
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: Oslo | Tel: (home): 47 22 49 56 53
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: //www.ifi.uio.no/~matmod/Personell/Hisdal_Ellen.html
:
:
:
:
: Also See
J F Baldwin, J Lawry , T P Martin (1996) A Note on Probability / Possibility Consistency for Fuzzy Events, Proceedings of IPMU 96 (the conference of Information Processing and Management of Uncertainty), Granada, Spain
Regards
Jonathan Lawry:
wq
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