Quantitative Possibility Theory and its Probabilistic Connections

Possibility theory is a representation framework general enough to model various kinds of information items: numbers, intervals, consonant random sets, special kind of probability families, as well as linguistic information, and uncertain formulae in logical settings. This paper focuses on quantitative possibility measures cast in the setting of imprecise probabilities. Recent results on possibility/probability transformations are recalled. The probabilistic interpretation of possibility measures sheds some light on defuzzification methods and suggests a common framework for fuzzy interval analysis and calculations with random parameters.

[1]  Radko Mesiar,et al.  Conditioning in possibility theory with strict order norms , 1999, Fuzzy Sets Syst..

[2]  Marc Roubens,et al.  Ranking and defuzzification methods based on area compensation , 1996, Fuzzy Sets Syst..

[3]  D. Dubois,et al.  A Semantics for Possibility Theory Based on Likelihoods , 1997 .

[4]  Didier Dubois,et al.  Consonant approximations of belief functions , 1990, Int. J. Approx. Reason..

[5]  E. Trillas,et al.  in Fuzzy Logic , 2002 .

[6]  Didier Dubois,et al.  Random sets and fuzzy interval analysis , 1991 .

[7]  George J. Klir,et al.  A MATHEMATICAL ANALYSIS OF INFORMATION-PRESERVING TRANSFORMATIONS BETWEEN PROBABILISTIC AND POSSIBILISTIC FORMULATIONS OF UNCERTAINTY , 1992 .

[8]  I. Hacking.  All Kinds of Possibility , 1975 .

[9]  Stanislaw Heilpern,et al.  Representation and application of fuzzy numbers , 1997, Fuzzy Sets Syst..

[10]  Andrew P. Sage,et al.  Uncertainty in Artificial Intelligence , 1987, IEEE Transactions on Systems, Man, and Cybernetics.

[11]  Wang Pei-zhuang.  From the Fuzzy Statistics to the Falling Random Subsets , 1983 .

[12]  Michael Clarke,et al.  Symbolic and Quantitative Approaches to Reasoning and Uncertainty , 1991, Lecture Notes in Computer Science.

[13]  D. Dubois,et al.  Unfair coins and necessity measures: Towards a possibilistic interpretation of histograms , 1983 .

[14]  Philippe Smets,et al.  Constructing the Pignistic Probability Function in a Context of Uncertainty , 1989, UAI.

[15]  Didier Dubois,et al.  Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty , 1996, Applied Intelligence.

[16]  Didier Dubois,et al.  Bayesian conditioning in possibility theory , 1997, Fuzzy Sets Syst..

[17]  D. Aeyels,et al.  On the Coherence of Supremum Preserving Upper Previsions , 1996 .

[18]  Rudolf Kruse,et al.  On an information compression view of possibility theory , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[19]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[20]  D. Dubois,et al.  On Possibility/Probability Transformations , 1993 .

[21]  Radko Mesiar,et al.  Fuzzy Interval Analysis , 2000 .

[22]  Dov M. Gabbay,et al.  Handbook of defeasible reasoning and uncertainty management systems: volume 2: reasoning with actual and potential contradictions , 1998 .

[23]  L. Shapley.  Cores of convex games , 1971 .

[24]  G. Cooman.  POSSIBILITY THEORY I: THE MEASURE- AND INTEGRAL-THEORETIC GROUNDWORK , 1997 .

[25]  Didier Dubois,et al.  Representing partial ignorance , 1996, IEEE Trans. Syst. Man Cybern. Part A.

[26]  George J. Klir,et al.  Fuzzy sets, uncertainty and information , 1988 .

[27]  Stanisław Heilpern,et al.  The expected value of a fuzzy number , 1992 .

[28]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[29]  Gert de Cooman.  Integration and Conditioning in Numerical Possibility Theory , 2004, Annals of Mathematics and Artificial Intelligence.

[30]  Cliff Joslyn,et al.  MEASUREMENT OF POSSIBILISTIC HISTOGRAMS FROM INTERVAL DATA , 1997 .

[31]  Didier Dubois,et al.  New Semantics for Quantitative Possibility Theory , 2001, ECSQARU.

[32]  M. Sugeno,et al.  Fuzzy measure of fuzzy events defined by fuzzy integrals , 1992 .

[33]  Vic Barnett,et al.  Comparative Statistical Inference , 1974, Technometrics.

[34]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[35]  George J. Klir,et al.  A principle of uncertainty and information invariance , 1990 .

[36]  Bernard Bobée,et al.  Revision of possibility distributions: A Bayesian inference pattern , 2000, Fuzzy Sets Syst..

[37]  G. Mauris,et al.  A fuzzy approach for the expression of uncertainty in measurement , 2001 .

[38]  J. J. Saade,et al.  Ordering fuzzy sets over the real line: an approach based on decision making under uncertainty , 1992 .

[39]  D. Dubois,et al.  When upper probabilities are possibility measures , 1992 .

[40]  Philippe Smets,et al.  The Transferable Belief Model , 1994, Artif. Intell..

[41]  Ronald R. Yager.  A Foundation for a Theory of Possibility , 1980, Cybern. Syst..

[42]  Didier Dubois,et al.  Possibility Theory: Qualitative and Quantitative Aspects , 1998 .

[43]  R. Yager.  On the specificity of a possibility distribution , 1992 .

[44]  D. Dubois,et al.  Fuzzy sets and statistical data , 1986 .

[45]  Etienne E. Kerre,et al.  Defuzzification: criteria and classification , 1999, Fuzzy Sets Syst..

[46]  Anne Lohrli.  Chapman and Hall , 1985 .

[47]  Ronald R. Yager,et al.  A procedure for ordering fuzzy subsets of the unit interval , 1981, Inf. Sci..

[48]  Gert de Cooman,et al.  A behavioral model for linguistic uncertainty , 2001, Inf. Sci..

[49]  G. Klir,et al.  PROBABILITY-POSSIBILITY TRANSFORMATIONS: A COMPARISON , 1992 .

[50]  James C. Bezdek,et al.  Analysis of fuzzy information , 1987 .

[51]  G. Klir,et al.  MEASURES OF UNCERTAINTY AND INFORMATION BASED ON POSSIBILITY DISTRIBUTIONS , 1982 .

[52]  S. Chanas,et al.  Single value simulation of fuzzy variables , 1988 .

[53]  Eric Raufaste,et al.  Empirical Evaluation of Possibility Theory in Human Radiological Diagnosis , 1998, ECAI.

[54]  Didier Dubois,et al.  A class of fuzzy measures based on triangular norms , 1982 .

[55]  Wolfgang Spohn,et al.  Ordinal Conditional Functions: A Dynamic Theory of Epistemic States , 1988 .

[56]  P. Walley.  Measures of Uncertainty in Expert Systems , 1996, Artificial Intelligence.

[57]  D. Dubois,et al.  The mean value of a fuzzy number , 1987 .

[58]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[59]  D Dubois,et al.  Belief structures, possibility theory and decomposable confidence measures on finite sets , 1986 .

[60]  Rudolf Kruse,et al.  The context model: An integrating view of vagueness and uncertainty , 1993, Int. J. Approx. Reason..

[61]  Paul P. Wang.  Advances in Fuzzy Sets, Possibility Theory, and Applications , 1983 .

[62]  S. Moral,et al.  On the concept of possibility-probability consistency , 1987 .

[63]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[64]  Lotfi A. Zadeh,et al.  A Theory of Approximate Reasoning , 1979 .

[65]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[66]  Hung T. Nguyen,et al.  Possibility Theory, Probability and Fuzzy Sets Misunderstandings, Bridges and Gaps , 2000 .

[67]  R. Yager,et al.  On the issue of defuzzification and selection based on a fuzzy set , 1993 .

[68]  David Lewis.  Counterfactuals and Comparative Possibility , 1973 .

[69]  E. Hisdal.  Conditional possibilities independence and noninteraction , 1978 .

[70]  María Ángeles Gil,et al.  A note on the connection between fuzzy numbers and random intervals , 1992 .

[71]  N. Shilkret.  Maxitive measure and integration , 1971 .

[72]  Didier Dubois,et al.  Evidence measures based on fuzzy information , 1985, Autom..

[73]  Gert de Cooman,et al.  Supremum Preserving Upper Probabilities , 1999, Inf. Sci..

[74]  Didier Dubois,et al.  Nonmonotonic Reasoning, Conditional Objects and Possibility Theory , 1997, Artif. Intell..

[75]  Antonio González.  A study of the ranking function approach through mean values , 1990 .