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The likelihood interpretation as the foundation of fuzzy set theory. (English) Zbl 1419.68157

Summary: In order to use fuzzy sets in real-world applications, an interpretation for the values of membership functions is needed. The history of fuzzy set theory shows that the interpretation in terms of statistical likelihood is very natural, although the connection between likelihood and probability can be misleading. In this paper, the likelihood interpretation of fuzzy sets is reviewed: it makes fuzzy data and fuzzy inferences perfectly compatible with standard statistical analyses, and sheds some light on the central role played by extension principle and \(\alpha\)-cuts in fuzzy set theory. Furthermore, the likelihood interpretation justifies some of the combination rules of fuzzy set theory, including the product and minimum rules for the conjunction of fuzzy sets, as well as the probabilistic-sum and bounded-sum rules for the disjunction of fuzzy sets.

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
03E72 Theory of fuzzy sets, etc.
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[1] Shafer, G., Belief functions and possibility measures, (Bezdek, J. C., Analysis of Fuzzy Information, vol. I: Mathematics and Logic (1987), CRC), 51-84
[2] Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman and Hall · Zbl 0732.62004
[3] Lindley, D. V., J. Am. Stat. Assoc., 99, 877-879 (2004), Comment to [58]
[4] Bradley, J., Fuzzy logic as a theory of vagueness: 15 conceptual questions, (Seising, R., Views on Fuzzy Sets and Systems from Different Perspectives (2009), Springer), 207-228 · Zbl 1173.03022
[5] Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning—I, Inf. Sci., 8, 199-249 (1975) · Zbl 0397.68071
[6] Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning—II, Inf. Sci., 8, 301-357 (1975) · Zbl 0404.68074
[7] Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning—III, Inf. Sci., 9, 43-80 (1975) · Zbl 0404.68075
[8] Watson, S. R.; Weiss, J. J.; Donnell, M. L., Fuzzy decision analysis, IEEE Trans. Syst. Man Cybern., 9, 1-9 (1979)
[9] Yager, R. R., Possibilistic decisionmaking, IEEE Trans. Syst. Man Cybern., 9, 388-392 (1979)
[10] Zadeh, L. A., Fuzzy probabilities, Inf. Process. Manag., 20, 363-372 (1984) · Zbl 0543.60007
[11] Puri, M. L.; Ralescu, D. A., Fuzzy random variables, J. Math. Anal. Appl., 114, 409-422 (1986) · Zbl 0592.60004
[12] Gil, M.Á.; Casals, M. R., An operative extension of the likelihood ratio test from fuzzy data, Stat. Pap., 29, 191-203 (1988) · Zbl 0671.62036
[13] Klir, G. J.; Yuan, B., Fuzzy Sets and Fuzzy Logic (1995), Prentice Hall · Zbl 0915.03001
[14] Kim, B.; Bishu, R. R., Evaluation of fuzzy linear regression models by comparing membership functions, Fuzzy Sets Syst., 100, 343-352 (1998)
[15] Inuiguchi, M.; Ramík, J., Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets Syst., 111, 3-28 (2000) · Zbl 0938.90074
[16] D’Urso, P., Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data, Comput. Stat. Data Anal., 42, 47-72 (2003) · Zbl 1429.62337
[17] Buckley, J. J., Fuzzy Probability and Statistics (2006), Springer · Zbl 1095.62002
[18] Gil, M.Á.; López-Díaz, M.; Ralescu, D. A., Overview on the development of fuzzy random variables, Fuzzy Sets Syst., 157, 2546-2557 (2006) · Zbl 1108.60006
[19] Nguyen, H. T.; Wu, B., Fundamentals of Statistics with Fuzzy Data (2006), Springer · Zbl 1100.62002
[20] Baudrit, C.; Couso, I.; Dubois, D., Joint propagation of probability and possibility in risk analysis: towards a formal framework, Int. J. Approx. Reason., 45, 82-105 (2007) · Zbl 1123.68123
[21] Choi, S. H.; Buckley, J. J., Fuzzy regression using least absolute deviation estimators, Soft Comput., 12, 257-263 (2008)
[22] Ferraro, M. B.; Coppi, R.; González-Rodríguez, G.; Colubi, A., A linear regression model for imprecise response, Int. J. Approx. Reason., 51, 759-770 (2010) · Zbl 1201.62086
[23] Denœux, T., Maximum likelihood estimation from fuzzy data using the EM algorithm, Fuzzy Sets Syst., 183, 72-91 (2011) · Zbl 1239.62017
[24] D’Urso, P.; Massari, R.; Santoro, A., Robust fuzzy regression analysis, Inf. Sci., 181, 4154-4174 (2011) · Zbl 1242.62073
[25] Viertl, R., Statistical Methods for Fuzzy Data (2011), Wiley · Zbl 1278.62014
[26] Georgescu, I., Possibility Theory and the Risk (2012), Springer · Zbl 1239.91002
[27] Wang, D.; Zhang, P.; Chen, L., Fuzzy fault tree analysis for fire and explosion of crude oil tanks, J. Loss Prev. Process Ind., 26, 1390-1398 (2013)
[28] Couso, I.; Dubois, D.; Sánchez, L., Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables (2014), Springer · Zbl 1355.60005
[29] Jung, H.-Y.; Lee, W.-J.; Yoon, J. H.; Choi, S. H., Likelihood inference based on fuzzy data in regression model, (SCIS & ISIS 2014 (2014), IEEE), 1175-1179
[30] Cattaneo, M., The likelihood interpretation of fuzzy data, (Ferraro, M. B.; Giordani, P.; Vantaggi, B.; Gagolewski, M.; Gil, M.Á.; Grzegorzewski, P.; Hryniewicz, O., Soft Methods for Data Science (2017), Springer), 113-120
[31] Seising, R., The Fuzzification of Systems (2007), Springer
[32] Fisher, R. A., On the “probable error” of a coefficient of correlation deduced from a small sample, Metron, 1, 3-32 (1921)
[33] Edwards, A. W.F., Likelihood (1992), Johns Hopkins University Press · Zbl 0833.62004
[34] Ford, J. L.; Ghose, S., The primitive uncertainty construct: possibility, potential surprise, probability and belief; some experimental evidence, Metroeconomica, 49, 195-220 (1998) · Zbl 0909.90004
[35] Pawitan, Y., In All Likelihood (2001), Oxford University Press · Zbl 1013.62001
[36] Cattaneo, M., Fuzzy probabilities based on the likelihood function, (Dubois, D.; Lubiano, M. A.; Prade, H.; Gil, M.Á.; Grzegorzewski, P.; Hryniewicz, O., Soft Methods for Handling Variability and Imprecision (2008), Springer), 43-50
[37] Raue, A.; Kreutz, C.; Maiwald, T.; Bachmann, J.; Schilling, M.; Klingmüller, U.; Timmer, J., Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood, Bioinformatics, 25, 1923-1929 (2009)
[38] Cattaneo, M., Likelihood decision functions, Electron. J. Stat., 7, 2924-2946 (2013) · Zbl 1280.62016
[39] Zadeh, L. A., Probability measures of fuzzy events, J. Math. Anal. Appl., 23, 421-427 (1968) · Zbl 0174.49002
[40] Liu, X.; Li, S., Cumulative distribution function estimation with fuzzy data: some estimators and further problems, (Kruse, R.; Berthold, M. R.; Moewes, C.; Gil, M.Á.; Grzegorzewski, P.; Hryniewicz, O., Synergies of Soft Computing and Statistics for Intelligent Data Analysis (2013), Springer), 83-91 · Zbl 1522.62024
[41] Heitjan, D. F.; Rubin, D. B., Ignorability and coarse data, Ann. Stat., 19, 2244-2253 (1991) · Zbl 0745.62004
[42] Little, R. J.A.; Rubin, D. B., Statistical Analysis with Missing Data (2002), Wiley · Zbl 1011.62004
[43] Carroll, R. J.; Ruppert, D.; Stefanski, L. A.; Crainiceanu, C. M., Measurement Error in Nonlinear Models (2006), Chapman & Hall/CRC
[44] Zadeh, L. A., Fuzzy sets, Inf. Control, 8, 338-353 (1965) · Zbl 0139.24606
[45] Gillies, D., Philosophical Theories of Probability (2000), Routledge
[46] Rowbottom, D. P., Probability (2015), Wiley
[47] Dubois, D.; Prade, H., Gradualness, uncertainty and bipolarity: making sense of fuzzy sets, Fuzzy Sets Syst., 192, 3-24 (2012) · Zbl 1238.03044
[48] Tversky, A.; Kahneman, D., Judgment under uncertainty: heuristics and biases, Science, 185, 1124-1131 (1974)
[49] Loginov, V. I., Probability treatment of Zadeh membership functions and their use in pattern recognition, Eng. Cybern., 4, 68-69 (1966)
[50] Black, M., Vagueness, Philos. Sci., 4, 427-455 (1937)
[51] Menger, K., Ensembles flous et fonctions aléatoires, C. R. Acad. Sci., 232, 2001-2003 (1951) · Zbl 0042.37202
[52] Hisdal, E., Are grades of membership probabilities?, Fuzzy Sets Syst., 25, 325-348 (1988) · Zbl 0664.04009
[53] Chen, Y. Y., Bernoulli trials: from a fuzzy measure point of view, J. Math. Anal. Appl., 175, 392-404 (1993) · Zbl 0773.60003
[54] Dubois, D.; Moral, S.; Prade, H., A semantics for possibility theory based on likelihoods, J. Math. Anal. Appl., 205, 359-380 (1997) · Zbl 0884.03017
[55] Bilgiç, T.; Türkşen, I. B., Measurement of membership functions: theoretical and empirical work, (Dubois, D.; Prade, H., Fundamentals of Fuzzy Sets (2000), Springer), 195-230 · Zbl 0974.03046
[56] Dubois, D.; Nguyen, H. T.; Prade, H., Possibility theory, probability and fuzzy sets, (Dubois, D.; Prade, H., Fundamentals of Fuzzy Sets (2000), Springer), 343-438 · Zbl 0978.94052
[57] Coletti, G.; Scozzafava, R., Conditional probability, fuzzy sets, and possibility: a unifying view, Fuzzy Sets Syst., 144, 227-249 (2004) · Zbl 1076.68082
[58] Singpurwalla, N. D.; Booker, J. M., Membership functions and probability measures of fuzzy sets, J. Am. Stat. Assoc., 99, 867-877 (2004) · Zbl 1117.62425
[59] Dubois, D., Possibility theory and statistical reasoning, Comput. Stat. Data Anal., 51, 47-69 (2006) · Zbl 1157.62309
[60] Coletti, G.; Vantaggi, B., From comparative degrees of belief to conditional measures, (Greco, S.; Marques Pereira, R. A.; Squillante, M.; Yager, R. R.; Kacprzyk, J., Preferences and Decisions (2010), Springer), 69-84 · Zbl 1205.60010
[61] Coletti, G.; Gervasi, O.; Tasso, S.; Vantaggi, B., Generalized Bayesian inference in a fuzzy context: from theory to a virtual reality application, Comput. Stat. Data Anal., 56, 967-980 (2012) · Zbl 1241.62022
[62] Coletti, G.; Vantaggi, B., Inference with probabilistic and fuzzy information, (Seising, R.; Trillas, E.; Moraga, C.; Termini, S., On Fuzziness, vol. 1 (2013), Springer), 115-119
[63] Scozzafava, R., The membership of a fuzzy set as coherent conditional probability, (Seising, R.; Trillas, E.; Moraga, C.; Termini, S., On Fuzziness, vol. 2 (2013), Springer), 631-635
[64] Kovalerchuk, B., Probabilistic solution of Zadeh’s test problems, (Laurent, A.; Strauss, O.; Bouchon-Meunier, B.; Yager, R. R., Information Processing and Management of Uncertainty in Knowledge-Based Systems, vol. 2 (2014), Springer), 536-545 · Zbl 1418.68205
[65] Edwards, A. W.F., The history of likelihood, Int. Stat. Rev., 42, 9-15 (1974) · Zbl 0289.62006
[66] Hald, A., On the history of maximum likelihood in relation to inverse probability and least squares, Stat. Sci., 14, 214-222 (1999) · Zbl 1059.62502
[67] Nguyen, H. T., On modeling of linguistic information using random sets, Inf. Sci., 34, 265-274 (1984) · Zbl 0557.68066
[68] De Cooman, G., Integration in possibility theory, (Grabisch, M.; Murofushi, T.; Sugeno, M., Fuzzy Measures and Integrals (2000), Physica-Verlag), 124-160 · Zbl 0957.28015
[69] Geyer, C. J.; Meeden, G. D., Fuzzy and randomized confidence intervals and \(P\)-values, Stat. Sci., 20, 358-366 (2005) · Zbl 1130.62319
[70] Walley, P., Measures of uncertainty in expert systems, Artif. Intell., 83, 1-58 (1996) · Zbl 1506.68157
[71] Cattaneo, M., On maxitive integration, Fuzzy Sets Syst., 304, 65-81 (2016) · Zbl 1368.28006
[72] Robert, C. P., The Bayesian Choice (2001), Springer
[73] Hryniewicz, O., Comparison of fuzzy and crisp random variables by Monte Carlo simulations, (Grzegorzewski, P.; Gagolewski, M.; Hryniewicz, O.; Gil, M.Á., Strengthening Links Between Data Analysis and Soft Computing (2015), Springer), 13-20 · Zbl 1385.60004
[74] Cattaneo, M.; Wiencierz, A., Likelihood-based imprecise regression, Int. J. Approx. Reason., 53, 1137-1154 (2012) · Zbl 1316.62116
[75] Nguyen, H. T., A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64, 369-380 (1978) · Zbl 0377.04004
[76] Mauris, G., Inferring a possibility distribution from very few measurements, (Dubois, D.; Lubiano, M. A.; Prade, H.; Gil, M.Á.; Grzegorzewski, P.; Hryniewicz, O., Soft Methods for Handling Variability and Imprecision (2008), Springer), 92-99
[77] Wilks, S. S., The large-sample distribution of the likelihood ratio for testing composite hypotheses, Ann. Math. Stat., 9, 60-62 (1938) · Zbl 0018.32003
[78] Cattaneo, M., A generalization of credal networks, (Augustin, T.; Coolen, F. P.A.; Moral, S.; Troffaes, M. C.M., ISIPTA ’09 (2009), SIPTA), 79-88
[79] Nau, R. F., Indeterminate probabilities on finite sets, Ann. Stat., 20, 1737-1767 (1992) · Zbl 0782.62006
[80] Walley, P., Statistical inferences based on a second-order possibility distribution, Int. J. Gen. Syst., 26, 337-383 (1997) · Zbl 0905.62026
[81] De Cooman, G.; Walley, P., A possibilistic hierarchical model for behaviour under uncertainty, Theory Decis., 52, 327-374 (2002) · Zbl 1148.91308
[82] De Cooman, G., A behavioural model for vague probability assessments, Fuzzy Sets Syst., 154, 305-358 (2005) · Zbl 1123.62006
[83] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE Trans. Autom. Control, 26, 926-936 (1981) · Zbl 1457.68262
[84] Dijkman, J. G.; van Haeringen, H.; de Lange, S. J., Fuzzy numbers, J. Math. Anal. Appl., 92, 301-341 (1983) · Zbl 0518.04004
[85] Fullér, R., On product-sum of triangular fuzzy numbers, Fuzzy Sets Syst., 41, 83-87 (1991) · Zbl 0725.04002
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