Skip to main content
SpringerLink
Log in

Quantitative Possibility Theory and its Probabilistic Connections

  • Conference paper
Soft Methods in Probability, Statistics and Data Analysis

Part of the book series: Advances in Intelligent and Soft Computing ((AINSC,volume 16))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barnett V. (1973). Comparative Statistical Inference, J. Wiley, New York.

    MATH  Google Scholar 

  2. Benferhat S., Dubois D. and Prade H. (1997a). Nonmonotonic reasoning, conditional objects and possibility theory, Artificial Intelligence, 92, 259–276.

    Article  MathSciNet  MATH  Google Scholar 

  3. Chanas S. and Nowakowski M. (1988). Single value simulation of fuzzy variable, Fuzzy Sets and Systems, 25, 43–57.

    Article  MathSciNet  MATH  Google Scholar 

  4. De Baets B., Tsiporkova E. and Mesiar R. (1999). Conditioning in possibility with strict order norms, Fuzzy Sets and Systems, 106, 221–229.

    Article  MathSciNet  MATH  Google Scholar 

  5. De Cooman G. (1997). Possibility theory–Part I: Measure-and integral-theoretics groundwork; Part II: Conditional possibility; Part III: Possibilistic independence, Int. J. of General Systems, 25 (4), 291–371.

    Article  MATH  Google Scholar 

  6. De Cooman G. (2001) Integration and conditioning in numerical possibility theory. Annals of Math. and AI, 32, 87–123.

    Google Scholar 

  7. De Cooman G. and Aeyels D. (1996). On the coherence of supremum preserving upper previsions, Proc. of the 6th Inter. Conf. Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’96), Granada, 1405–1410.

    Google Scholar 

  8. De Cooman G., Aeyels D. (1999). Supremum-preserving upper probabilities. Information Sciences, 118, 173–212.

    Article  MathSciNet  MATH  Google Scholar 

  9. Delgado M. and Moral S. (1987). On the concept of possibility-probability consistency, Fuzzy Sets and Systems, 21, 311–318.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dempster A. P. (1967). Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Stat., 38, 325–339.

    Article  MathSciNet  MATH  Google Scholar 

  11. Denneberg D. (1994). Nonadditive Measure and Integral, Kluwer Academic, Dordrecht, The Netherlands.

    Google Scholar 

  12. Dubois D. (1986). Belief structures, possibility theory and decomposable confidence measures on finite sets, Computers and Artificial Intelligence (Bratislava), 5, 403–416.

    MATH  Google Scholar 

  13. Dubois D., Fargier H; and Prade H. (1996b). Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty, Applied Intelligence, 6, 287–309.

    Article  Google Scholar 

  14. Dubois D., Kerre E., Mesiar R., Prade H. Fuzzy interval analysis. In: Fundamentals of Fuzzy Sets, Dubois,D. Prade,H., Eds: Kluwer, Boston, Mass, The Handbooks of Fuzzy Sets Series, 483–581, 2000.

    Google Scholar 

  15. Dubois D., Moral S. and Prade H. (1997). A semantics for possibility theory based on likelihoods, J. of Mathematical Analysis and Applications, 205, 359–380.

    Article  MathSciNet  MATH  Google Scholar 

  16. Dubois D. and Prade H. (1980). Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York.

    MATH  Google Scholar 

  17. Dubois and Prade H. (1982) On several representations of an uncertain body of evidence,“ in Fuzzy Information and Decision Processes, M.M. Gupta, and E. Sanchez, Eds., North-Holland, Amsterdam, 1982, pp. 167–181.

    Google Scholar 

  18. Dubois D. and Prade H. (1983) Unfair coins and necessity measures: towards a possibilistic interpretation of histograms. Fuzzy Sets and Systems, 10, 15–20.

    Article  MathSciNet  MATH  Google Scholar 

  19. Dubois D. and Prade H. (1985). Evidence measures based on fuzzy information, Automatica, 21, 547–562.

    Article  MathSciNet  MATH  Google Scholar 

  20. Dubois D. and Prade H. (1986). Fuzzy sets and statistical data, Europ. J. Operations Research, 25, 345–356.

    Article  MathSciNet  MATH  Google Scholar 

  21. Dubois D. and Prade H. (1987). The mean value of a fuzzy number, Fuzzy Sets and Systems, 24, 279–300.

    Article  MathSciNet  MATH  Google Scholar 

  22. Dubois D. and Prade H. (1988). Possibility Theory, Plenum Press, New York.

    Book  Google Scholar 

  23. Dubois D. and Prade H. (1990) Consonant approximations of belief functions. Int. J. Approximate Reasoning, 4, 419–449.

    Article  MathSciNet  MATH  Google Scholar 

  24. Dubois D. and Prade H. (1991). Random sets and fuzzy interval analysis, Fuzzy Sets and Systems, 42, 87–101.

    Article  MathSciNet  MATH  Google Scholar 

  25. Dubois D. and Prade H. (1992). When upper probabilities are possibility measures, Fuzzy Sets and Systems, 49, 65–74.

    Article  MathSciNet  MATH  Google Scholar 

  26. Dubois D. and Prade H. (1997) Bayesian conditioning in possibility theory, Fuzzy Sets and Systems, 92, 223–240.

    Article  MathSciNet  MATH  Google Scholar 

  27. Dubois D. and Prade H. (1998). Possibility theory: Qualitative and quantitative aspects, Handbook of Defeasible Reasoning and Uncertainty Management Systems–Vol. 1 ( Gabbay D.M. and Smets P., eds.), Kluwer Academic Publ., Dordrecht, 169–226.

    Google Scholar 

  28. Dubois D., Nguyen H. T., Prade H. (2000) Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps. In: Fundamentals of Fuzzy Sets, (Dubois, D. Prade,H., Eds.), Kluwer, Boston, Mass., The Handbooks of Fuzzy Sets Series, 343–438.

    Google Scholar 

  29. Dubois D., Foulloy L., Mauris G., Prade H. (2002), Possibility/probability transformations, triangular fuzzy sets, and probabilistic inequalities. Proc. IPMU conference, Annecy, France.

    Google Scholar 

  30. Dubois D., Prade H. and Sandri S. (1993). On possibility/probability transformations. In: Fuzzy Logic. State of the Art, ( R. Lowen, M. Roubens, eds.), Kluwer Acad. Publ., Dordrecht, 103–112.

    Chapter  Google Scholar 

  31. Dubois D., Prade H. and Smets P. (1996). Representing partial ignorance, IEEE Trans. on Systems, Man and Cybernetics, 26, 361–377.

    Article  Google Scholar 

  32. Dubois D., Prade H. and Smets P. (2001). New semantics for quantitative possibility theory. Proc. ESQARU 2001, Toulouse, LNAI 2143, Springer-Verlag, p. 410–421, 19–21.

    Google Scholar 

  33. Edwards W. F. (1972). Likelihood, Cambridge University Press, Cambridge, U.K.

    MATH  Google Scholar 

  34. Fortemps P. and Roubens M. (1996). Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems, 82, 319–330.

    Article  MathSciNet  MATH  Google Scholar 

  35. Gebhardt J. and Kruse R. (1993). The context model–an intergating view of vagueness and uncertainty. Int. J. Approximate Reasoning, 9, 283–314.

    Article  MathSciNet  MATH  Google Scholar 

  36. Gebhardt J. and Kruse R. (1994a) A new approach to semantic aspects of possibilistic reasoning. Symbolic and Quantitative Approaches to Reasoning and Uncertainty (M. Clarke et al. Eds.), Lecture Notes in Computer Sciences Vol. 747, Springer Verlag, 151–160.

    Google Scholar 

  37. Gebhardt J. and Kruse R. (1994b) On an information compression view of possibility theory. Proc 3rd IEEE Int. Conference on Fuzzy Systems. Orlando,Fl., 1285–1288.

    Google Scholar 

  38. Geer J.F. and Klir G.J. (1992). A mathematical analysis of information-preserving transformations between probabilistic and possibilistic formulations of uncertainty, Int. J. of General Systems, 20, 143–176.

    Article  MATH  Google Scholar 

  39. Gil M. A. (1992). A note on the connection between fuzzy numbers and random intervals, Statistics and Probability Lett., 13, 311–319.

    Article  MATH  Google Scholar 

  40. Gonzalez A. (1990). A study of the ranking function approach through mean values, Fuzzy Sets and Systems, 35, 29–43.

    Article  MathSciNet  MATH  Google Scholar 

  41. Grabisch M., Murofushi T. and Sugeno M. (1992). Fuzzy measure of fuzzy events defined by fuzzy integrals, Fuzzy Sets and Systems, 50, 293–313.

    Article  MathSciNet  MATH  Google Scholar 

  42. Hacking I. (1975). All kinds of possibility, Philosophical Review, 84, 321–347.

    Article  Google Scholar 

  43. Heilpern S. (1992). The expected value of a fuzzy number, Fuzzy Sets and Systems, 47, 81–87.

    Article  MathSciNet  MATH  Google Scholar 

  44. Heilpern S. (1997). Representation and application of fuzzy numbers, Fuzzy Sets and Systems, 91, 259–268.

    Article  MathSciNet  MATH  Google Scholar 

  45. Higashi and Klir G. (1982). Measures of uncertainty and information based on possibility distributions, Int. J. General Systems, 8, 43–58.

    Article  Google Scholar 

  46. Hisdal E. (1978). Conditional possibilities independence and noninteraction, Fuzzy Sets and Systems, 1, 283–297.

    Article  MATH  Google Scholar 

  47. Joslyn C. (1997). Measurement of possibilistic histograms from interval data, Int. J. of General Systems, 26 (1–2), 9–33.

    Article  MathSciNet  MATH  Google Scholar 

  48. Kaufmann A (1980). La simulation des ensembles flous, CNRS Round Table on Fuzzy Sets, Lyon, France (unpublished proceedings).

    Google Scholar 

  49. Klir G.J. (1990). A principle of uncertainty and information invariance, Int. J. of General Systems, 17, 249–275.

    Article  MATH  Google Scholar 

  50. Klir G.J. and Folger T. (1988). Fuzzy Sets, Uncertainty and Information, Prentice Hall, Englewood Cliffs, NJ.

    Google Scholar 

  51. Klir G.J. and Parviz B. (1992). Probability-possibility transformations: A comparison, Int. J. of General Systems, 21, 291–310.

    Article  MATH  Google Scholar 

  52. Lapointe S. Bobee B. (2000). Revision of possibility distributions: A Bayesian inference pattern, Fuzzy Sets and Systems, 116, 119–140

    Article  MathSciNet  MATH  Google Scholar 

  53. Mauris G., Lasserre V., Foulloy L. (2001). A fuzzy appproach for the expression of uncertainty in measurement. Int. J. Measurement, 29, 165–177.

    Google Scholar 

  54. Lewis D. L. (1979). Counterfactuals and comparative possibility, Ifs (Harper W. L., Stalnaker R. and Pearce G., eds.), D. Reidel, Dordrecht, 57–86.

    Google Scholar 

  55. Raufaste E. and Da Silva Neves R. (1998). Empirical evaluation of possibility theory in human radiological diagnosis, Proc. of the 13th Europ. Conf. on Artificial Intelligence (ECAI’98) ( Prade H., ed. ), John Wiley amp; Sons, 124–128.

    Google Scholar 

  56. Saade J. J. and Schwarzlander H. (1992). Ordering fuzzy sets over the real line: An approach based on decision making under uncertainty, Fuzzy Sets and Systems, 50, 237–246.

    Article  MathSciNet  Google Scholar 

  57. Shackle G. L.S. (1961). Decision, Order and Time in Human Affairs, ( 2nd edition ), Cambridge University Press, UK.

    Google Scholar 

  58. Shafer G. (1976). A Mathematical Theory of Evidence, Princeton University Press, Princeton.

    Google Scholar 

  59. Shafer G. (1987). Belief functions and possibility measures, Analysis of Fuzzy Information–Vol. I: Mathematics and Logic (Bezdek J.C., Ed.), CRC Press, Boca Raton, FL, 51–84.

    Google Scholar 

  60. Shapley S. (1971). Cores of convex games, Int. J. of Game Theory, 1, 12–26.

    MathSciNet  Google Scholar 

  61. Shilkret N. (1971). Maxitive measure and integration, Indag. Math., 33, 109116.

    Google Scholar 

  62. Smets P. (1990). Constructing the pignistic probability function in a context of uncertainty, Uncertainty in Artificial Intelligence 5 (Henrion M. et al., Eds.), North-Holland, Amsterdam, 29–39.

    Google Scholar 

  63. Smets P. and Kennes R. (1994). The transferable belief model, Artificial Intelligence, 66. 191–234.

    Article  MathSciNet  Google Scholar 

  64. Spohn W. (1988). Ordinal conditional functions: A dynamic theory of epistemic states, Causation in Decision, Belief Change and Statistics (Harper W. and Skyrms B., Eds. ), 105–134.

    Book  Google Scholar 

  65. Van Leekwijk W. and Kerre E. (1999). Defuzzification: criteria and classification. Fuzzy Sets and Systems., 118, 159–178.

    Article  Google Scholar 

  66. Walley P. (1991). Statistical Reasoning with Imprecise Probabilities, Chapman and Hall.

    Google Scholar 

  67. Walley P. (1996). Measures of uncertainty in expert systems, Artificial Intelligence, 83, 1–58.

    Article  MathSciNet  Google Scholar 

  68. Walley P. and de Cooman G. (1999) A behavioural model for linguistic uncertainty, Information Sciences, 134, 1–37.

    Article  Google Scholar 

  69. Wang P.Z. (1983). From the fuzzy statistics to the falling random subsets, Advances in Fuzzy Sets, Possibility Theory and Applications (Wang P.P., Eds.), Plenum Press, New York, 81–96.

    Book  Google Scholar 

  70. Yager R.R. (1980). A foundation for a theory of possibility, J. Cybernetics, 10, 177–204.

    Article  MathSciNet  MATH  Google Scholar 

  71. Yager R. R. (1981). A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, 24, 143–161.

    Article  MathSciNet  MATH  Google Scholar 

  72. Yager R.R. (1992). On the specificity of a possibility distribution, Fuzzy Sets and Systems, 50, 279–292.

    Article  MathSciNet  MATH  Google Scholar 

  73. Yager R. R. and Filev D. (1993). On the issue of defuzzification and selection based on a fuzzy set, Fuzzy Sets and Systems, 55, 255–271.

    Article  MathSciNet  MATH  Google Scholar 

  74. Zadeh L.A. (1965). Fuzzy sets, Information and Control, 8, 338–353.

    Article  MathSciNet  MATH  Google Scholar 

  75. Zadeh L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning, Information Sciences, Part I: 8, 199–249; Part II: 8, 301–357; Part III: 9, 43–80.

    MathSciNet  MATH  Google Scholar 

  76. Zadeh L. A. (1978). Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3–28.

    Article  MathSciNet  MATH  Google Scholar 

  77. Zadeh L.A. (1979a). A theory of approximate reasoning, Machine Intelligence, Vol. 9 ( Hayes J. E., Michie D. and Mikulich L. I., eds.), John Wiley amp; Sons, New York, 149–194.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRIT-CNRS, Universite Paul Sabatier, Toulouse, France

    Didier Dubois & Henri Prade

Authors
  1. Didier Dubois
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henri Prade
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Przemysław Grzegorzewski  & Olgierd Hryniewicz  & 

  2. Facultad de Ciencias, Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, C/Calvo Sotelo, s/n, 33007, Oviedo, Spain

    María Ángeles Gil

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dubois, D., Prade, H. (2002). Quantitative Possibility Theory and its Probabilistic Connections. In: Grzegorzewski, P., Hryniewicz, O., Gil, M.Á. (eds) Soft Methods in Probability, Statistics and Data Analysis. Advances in Intelligent and Soft Computing, vol 16. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1773-7_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-7908-1773-7_1

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-1526-9

  • Online ISBN: 978-3-7908-1773-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation