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Coherent Conditional Plausibility: A Tool for Handling Fuzziness and Uncertainty Under Partial Information

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Soft Computing Applications for Group Decision-making and Consensus Modeling

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 357))

Abstract

Non-additive measures, such as plausibility, are meaningful when only a partial or indirect information on the events of interest is available, or when imprecision and ambiguity of agents are considered. Our main aim is to study inferential processes, like the Bayesian one, when the information is expressed in natural language and the uncertainty measure is either partially or imprecisely assessed. We deal with partial assessments consistent with a conditional plausibility, and adopt the interpretation of the membership of a fuzzy set in terms of coherent conditional plausibility, regarded as a function of the conditioning events. This kind of interpretation, inspired to that given in terms of coherent conditional probability, is particularly useful for computing the measure of the uncertainty of fuzzy events, when the knowledge on the variable is imprecise and can be managed with a non-additive measure of uncertainty. A simple situation related to a Zadeh’s example can be the following: a ball will be drawn from an urn containing balls of different colours and different diameters, but one knows only the distribution of the different colours. The aim is to compute the uncertainty measure of the fuzzy event “a small ball is drawn”, taking in considerations possible logical constrains among the colours and the diameters.

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References

  1. Baioletti M, Coletti G, Petturiti D, Vantaggi B (2011) Inferential models and relevant algorithms in a possibilistic framework. Int J Approx Reason 52:580–598

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouchon-Meunier B, Coletti G, Marsala C (2002) Independence and possibilistic conditioning. Ann Math Artif Intell 35:107–123

    Article  MathSciNet  MATH  Google Scholar 

  3. Brunelli M, Fedrizzi M (2010) The representation of uncertainty in operational research: considerations on the use of possibility, probability, and fuzzy numbers. In: Proceedings of the 2nd international conference on applied operational research, Uniprint, Turku, pp 24–33. ISBN 9789521224140

    Google Scholar 

  4. Capotorti A, Coletti G, Vantaggi B (2014) Standard and nonstandard representability of positive uncertainty orderings. Kybertika 50:189–215

    MathSciNet  MATH  Google Scholar 

  5. Chateauneuf A, Jaffray JY (1989) Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion. Math Soc Sci 17(3):263–283

    Article  MATH  Google Scholar 

  6. Coletti G, Gervasi O, Tasso S, Vantaggi B (2012) Generalized Bayesian inference in a fuzzy context: from theory to a virtual reality application. Comput Stat Data Anal 56:967–980

    Google Scholar 

  7. Coletti G, Petturiti D, Vantaggi B (2016) Conditional belief functions as lower envelopes of conditional probabilities in a finite setting. Inf Sci 339:64–84

    Article  MathSciNet  Google Scholar 

  8. Coletti G, Petturiti D, Vantaggi B (2014) Possibilistic and probabilistic likelihood functions and their extensions: common features and specific characteristics. Fuzzy Sets Syst 250:25–51

    Article  MathSciNet  MATH  Google Scholar 

  9. Coletti G, Petturiti D, Vantaggi B (2014) Possibilistic and probabilistic likelihood functions and their extensions: common features and specific characteristics. Fuzzy Sets Syst 250:25–51

    Article  MathSciNet  MATH  Google Scholar 

  10. Coletti G, Petturiti D, Vantaggi B (in press) Fuzzy memberships as likelihood functions in a possibilistic framework. Int J Approx Reason. doi:10.1016/j.ijar.2016.11.017

  11. Coletti G, Scozzafava R (2002) Probabilistic logic in a coherent setting. Trends in logic, no 15. Kluwer, Dordrecht

    Google Scholar 

  12. Coletti G, Scozzafava R (2004) Conditional probability, fuzzy sets, and possibility: a unifying view. Fuzzy Sets Syst 144:227–249

    Article  MathSciNet  MATH  Google Scholar 

  13. Coletti G, Scozzafava R (2006) Conditional probability and fuzzy information. Comput Stat Data Anal 51:115–132

    Article  MathSciNet  MATH  Google Scholar 

  14. Coletti G, Scozzafava R, Vantaggi B (2013) Inferential processes leading to possibility and necessity. Inf Sci 245:132–145

    Article  MathSciNet  MATH  Google Scholar 

  15. Coletti G, Scozzafava R (2006) Toward a general theory of conditional beliefs. Int J Intell Syst 21:229–259

    Article  MATH  Google Scholar 

  16. Coletti G, Scozzafava R, Vantaggi B (2009) Integrated likelihood in a finitely additive setting. Lect Notes Comput Sci: LNAI 5590:554–565

    Article  MathSciNet  MATH  Google Scholar 

  17. Coletti G, Vantaggi B (2006) Possibility theory: conditional independence. Fuzzy Sets Syst 157(11):1491–1513

    Article  MathSciNet  MATH  Google Scholar 

  18. Coletti G, Vantaggi B (2009) T-conditional possibilities: coherence and inference. Fuzzy Set Syst 160:306–324

    Article  MathSciNet  MATH  Google Scholar 

  19. Coletti G, Vantaggi B (2012) Probabilistic reasoning in a fuzzy context. In: Proceedings of the second world conference on soft computing, Baku, Azerbaijan, pp 65–72

    Google Scholar 

  20. Coletti G, Vantaggi B (2013) Hybrid models: probabilistic and fuzzy information. Synergies of soft computing and statistics for intelligent data analysis. In: Kruse R, Berthold MR, Moewes C, Gil MA, Grzegorzewski P, Hryniewicz O (eds) Advances in intelligent systems and computing, pp 63–72

    Google Scholar 

  21. Coletti G, Vantaggi B (2013) Conditional nonadditive measures and fuzzy sets. In: Proceedings of 8th international symposium on imprecise probability: theories and applications, Compiégne, France, 2013 (ISIPTA 2013)

    Google Scholar 

  22. Cuzzolin F (2010) Three alternative combinatorial formulations of the theory of evidence. J Intell Data Anal 14:439–464

    Google Scholar 

  23. de Finetti B (1970) Teoria della probabilitá. Einaudi, Torino (Engl. Transl. (1974) Theory of probability vols I, II. Wiley, London)

    Google Scholar 

  24. de Finetti B (1949) Sull’impostazione assiomatica del calcolo delle probabilità. Annali Univ. Trieste 19:3–55. (Engl. transl.: Ch. 5 in Probability, induction, statistics. Wiley, London (1972))

    Google Scholar 

  25. de Cooman G, Miranda E, Couso I (2005) Lower previsions induced by multi-valued mappings. J Stat Plan Infer 133:173–197

    Article  MathSciNet  MATH  Google Scholar 

  26. Dempster AP (1968) A generalization of Bayesian inference. R Stat Soc B 50:205–247

    MathSciNet  MATH  Google Scholar 

  27. Brüning M, Denneberg D (2002) Max-min (\(\sigma \)-)additive representation of monotone measures. Stat Pap 43(1):23–35

    Article  MathSciNet  MATH  Google Scholar 

  28. Denoeux T, Smets P (2006) Classification using belief functions: the relationship between the case-based and model-based approaches. IEEE Trans Syst Man Cybern B 36(6):1395–1406

    Article  Google Scholar 

  29. Dubins L (1975) Finitely additive conditional probabilities, conglomerability and disintegrations. Ann Prob 3(1):89–99

    Article  MathSciNet  MATH  Google Scholar 

  30. Dubois D, Prade H (1986) New results about properties and semantics of fuzzy set-theoretic operators. In: Wang PP, Chang SK (eds) Fuzzy Sets. Plenum Press, New York, pp 59–75

    Google Scholar 

  31. Dubois D, Moral S, Prade H (1997) A semantics for possibility theory based on likelihoods. J Math Anal Appl 205:359–380

    Article  MathSciNet  MATH  Google Scholar 

  32. Dubois D, Prade H (1989) Fuzzy sets, probability and measurement. Eur J Oper Res 40:135–154

    Google Scholar 

  33. Dubois D, Prade H (1992) When upper probabilities are possibility measures. Fuzzy Sets Syst 49:65–74

    Article  MathSciNet  MATH  Google Scholar 

  34. Fagin R, Halpern JY (1991) A new approach to updating beliefs. In: Bonissone PP, Henrion M, Kanal LN, Lemmer JF (eds) Uncertainty in artificial intelligence, vol 6, pp 347–374

    Google Scholar 

  35. Fedrizzi M, Esogbue A, Kacprzyk J (1988) Fuzzy dynamic programming with stochastic systems. In: Kacprzyk J, Fedrizzi M (a cura di) Combining fuzzy imprecision with probabilistic uncertainty in decision making. Lecture notes in economics and mathematical systems. Springer, Berlin, pp 266–285

    Google Scholar 

  36. Fedrizzi M, Kacprzyk J (a cura di) (1988) Combining fuzzy imprecision with probabilistic uncertainty in decision making. Lecture notes in economics and mathematical systems, vol 310. Springer, Berlin, 399 pp

    Google Scholar 

  37. Frank MJ (1979) On the simultaneous associativity of \(F(x, y)\) and \(x+y-F(x, y)\). Aequationes Math. 19:194–226

    Article  MathSciNet  MATH  Google Scholar 

  38. Halpern J (2003) Reasoning about uncertainty. The MIT Press, Boston

    MATH  Google Scholar 

  39. Hisdal E (1988) Are grades of membership probabilities. Fuzzy Sets Syst 25:325–348

    Article  MathSciNet  MATH  Google Scholar 

  40. Jaffray JY (1992) Bayesian updating and belief functions. IEEE Trans Syst Man Cybern 22:1144–1152

    Article  MathSciNet  MATH  Google Scholar 

  41. Krauss PH (1968) Representation of conditional probability measures on Boolean algebras. Acta Mathematica Academiae Scientiarum Hungaricae 19(3–4):229–241

    Article  MathSciNet  MATH  Google Scholar 

  42. Liu Z, Li HX (2005) A probabilistic fuzzy logic system for modeling and control. IEEE Trans Fuzzy Syst 13(6):848–859

    Article  Google Scholar 

  43. Loginov VI (1966) Probability treatment of Zadeh membership functions and theris use in patter recognition. Eng Cybern 68–69

    Google Scholar 

  44. Lukasiewicz T, Straccia U (2008) Managing uncertainty and vagueness in description logics for the Semantic Web. Web Semant: Sci Serv Agents World Wide Web 6(4):291–308

    Article  Google Scholar 

  45. Mastroleo M, Vantaggi B (2007) An independence concept under plausibility function. In: Proceeding of 5th international symposium on imprecise probabilities and their applications, pp 287–296

    Google Scholar 

  46. Petturiti D, Vantaggi B (2017) Envelopes of conditional probabilities extending a strategy and a prior probability. Int J Approx Reason 81:160–182

    Article  MathSciNet  MATH  Google Scholar 

  47. Singpurwalla ND, Booker JM (2004) Membership functions and probability measures of fuzzy sets (with discussion). J Am Stat Assoc 99:867–889

    Article  MATH  Google Scholar 

  48. Tripathy BK, Mohant RK, Sooraj TR (2016) Application of uncertainty models in bioinformatics, chapter 9. In: Dash S, Subudhi B, Dash S (eds) Handbook of research on computational intelligence applications in bioinformatics. IGI Global, pp 169–182

    Google Scholar 

  49. Yager RR (1980) On a general class of fuzzy connectives. Fuzzy Sets Syst 4:235–242

    Article  MathSciNet  MATH  Google Scholar 

  50. Walley P (1987) Belief function representations of statistical evidence. Ann Stat 4:1439–1465

    Article  MathSciNet  MATH  Google Scholar 

  51. Walley P (1991) Statistical reasoning with imprecise probabilities. Chapman and Hall, London

    Book  MATH  Google Scholar 

  52. Weber S (1983) A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms. Fuzzy Sets Syst 11:115–134

    Article  MathSciNet  MATH  Google Scholar 

  53. Zadeh L (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MATH  Google Scholar 

  54. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427

    Article  MathSciNet  MATH  Google Scholar 

  55. Zadeh LA (2002) Toward a perception-based theory of probabilistic reasoning with imprecise probabilities. J Stat Plan Infer 105:233–264

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhao R, Liu B (2004) Redundancy optimization problems with uncertainty of combining randomness and fuzziness. Eur J Oper Res 157(3):716–735

    Article  MathSciNet  MATH  Google Scholar 

  57. Zimmermann H-J (1987) Fuzzy sets, decision making, and expert systems. Kluwer, Boston

    Google Scholar 

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Acknowledgements

This research was partially supported by by GNAMPA of INdAM.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli, 1, 06123, Perugia, Italy

    Giulianella Coletti

  2. Dipartimento S.B.A.I, “La Sapienza” Università di Roma, via Scarpa, 00185, Rome, Italy

    Barbara Vantaggi

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  1. Giulianella Coletti
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  2. Barbara Vantaggi
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Correspondence to Giulianella Coletti .

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Editors and Affiliations

  1. School of Business and Management, Lappeenranta University of Technology, Lappeenranta, Finland

    Mikael Collan

  2. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

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Coletti, G., Vantaggi, B. (2018). Coherent Conditional Plausibility: A Tool for Handling Fuzziness and Uncertainty Under Partial Information. In: Collan, M., Kacprzyk, J. (eds) Soft Computing Applications for Group Decision-making and Consensus Modeling. Studies in Fuzziness and Soft Computing, vol 357. Springer, Cham. https://doi.org/10.1007/978-3-319-60207-3_9

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  • DOI: https://doi.org/10.1007/978-3-319-60207-3_9

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-60207-3

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