Fuzzy Set Theory and Probability Theory: What is the Relationship?
Relationship between probability theory and fuzzy set theory is associated with a long history of discussion and debate. My first paper on fuzzy sets was published in 1965 (Zadeh 1965). In a paper published in 1966, Loginov suggested that the membership function of a fuzzy set may be interpreted as a conditional probability (Loginov 1966). Subsequently, related links to probability theory were suggested and analyzed by many others (Coletti and Scozzafava 2004; Freeling 1981; Hisdal 1986a, b; Nurmi 1977; Ross et al. 2002; Singpurwalla and Booker 2004; Stallings 1977; Thomas 1995; Viertl 1987; Yager 1984). Among such links are links to set-valued random variables (Goodmanm and Nguyen 1985; Orlov 1980; Wang and Sanchez 1982) and to the Dempster–Shafer theory (Dempster 1967; Shafer 1976). A more detailed discussion of these links may be found in my l995 paper “Probability theory and fuzzy logic are complementary rather...
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Coletti G, Scozzafava R (2004) Conditional probability, fuzzy sets, and possibility: a unifying view. Fuzzy Set Syst 144(1):227–249
Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38:325–329
Dubois D, Nguyen HT, Prade H (2000) Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps. In: Dubois D, Prade H (eds) Fundamentals of fuzzy sets. The handbooks of fuzzy sets series. Kluwer, Boston, MA, pp 343–438
Freeling ANS (1981) Possibilities versus fuzzy probabilities – Two alternative decision aids. Tech. Rep. 81–6, Decision Science Consortium Inc., Washington, DC
Goodman IR, Nguyen HT (1985) Uncertainty models for knowledge-based systems. North Holland, Amsterdam
Hisdal E (1986) Infinite-valued logic based on two-valued logic and probability. Part 1.1: Difficulties with present-day fuzzy-set theory and their resolution in the TEE model. Int J Man-Mach Stud 25(1):89–111
Hisdal E (1986) Infinite-valued logic based on two-valued logic and probability. Part 1.2: Different sources of fuzziness. Int J Man-Mach Stud 25(2):113–138
Lindley DV (1987) The probability approach to the treatment of uncertainty in artificial intelligence and expert systems. Statistical Science 2:17–24
Loginov VJ (1966) Probability treatment of Zadeh membership functions and their use in pattern recognition, Eng Cybern 68–69
Nurmi H (1977) Probability and fuzziness: some methodological considerations. Unpublished paper presented at the sixth research conference on subjective probability, utility, and decision making, Warszawa
Orlov AI (1980) Problems of optimization and fuzzy variables. Znaniye, Moscow
Ross TJ, Booker JM, Parkinson WJ (eds) (2002) Fuzzy logic and probability applications: bridging the gap. Society for Industrial and Applied Mathematics, Philadelphia, PA
Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton, NJ
Singpurwalla ND, Booker JM (2004) Membership functions and probability measures of fuzzy sets. J Am Stat Assoc 99:467
Stallings W (1977) Fuzzy set theory versus Bayesian statistics. IEEE Trans Syst Man Cybern, SMC-7:216–219
Thomas SF (1995) Fuzziness and probability, ACG Press, Wichita KS
Viertl R (1987) Is it necessary to develop a fuzzy Bayesian inference? In: Viertl R (ed) Probability and Bayesian statistics. Plenum, New York, pp 471–475
Wang PZ, Sanchez E (1982) Treating a fuzzy subset as a projectable random set. In: Gupta MM, Sanchez E (eds) Fuzzy information and decision processes. North Holland, Amsterdam, pp 213–220
Yager RR (1984) Probabilities from fuzzy observations. Inf Sci 32:1–31
Zadeh LA (1965) Fuzzy sets. Inform Contr 8:338–353
Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Set Syst 1:3–28
Zadeh LA (1986) Is probability theory sufficient for dealing with uncertainty in AI: a negative view. In: Kanal LN, Lemmer JF (eds) Uncertainty in artificial intelligence. North Holland, Amsterdam
Zadeh LA (1995) Probability theory and fuzzy logic are complementary rather than competitive. Technometrics 37:271–276
Zadeh LA (2002) Toward a perception-based theory of probabilistic reasoning with imprecise probabilities. J Stat Plan Inference 105:233–264
Zadeh LA (2004) Probability theory and fuzzy logic – a radical view. J Am Stat Assoc 99(467):880–881
Zadeh LA (2005) Toward a generalized theory of uncertainty (GTU) – an outline. Inf Sci 172:1–40
Zadeh LA (2006) Generalized theory of uncertainty (GTU) – principal concepts and ideas. Comput Stat Data Anal 51:15–46
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Zadeh, L.A. (2011). Fuzzy Set Theory and Probability Theory: What is the Relationship?. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_614
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