Article

Full entry | PDF   (0.3 MB)
Keywords:
probability measures; possibility measures; non-numerical uncertainty degrees; set-valued uncertainty degrees; possibilistic uncertainty functions; set-valued entropy functions
Summary:
When proposing and processing uncertainty decision-making algorithms of various kinds and purposes, we more and more often meet probability distributions ascribing non-numerical uncertainty degrees to random events. The reason is that we have to process systems of uncertainties for which the classical conditions like $\sigma$-additivity or linear ordering of values are too restrictive to define sufficiently closely the nature of uncertainty we would like to specify and process. In cases of non-numerical uncertainty degrees, at least the following two criteria may be considered. The first criterion should be systems with rather complicated, but sophisticated and nontrivially formally analyzable uncertainty degrees, e. g., uncertainties supported by some algebras or partially ordered structures. Contrarily, we may consider easier relations, which are non-numerical but interpretable on the intuitive level. Well-known examples of such structures are set-valued possibilistic measures. Some specific interesting results in this direction are introduced and analyzed in this contribution.
References:
[1] Birkhoff, G.: Lattice Theory. Third edition. Providence, Rhode Island 1967. MR 0227053
[2] DeCooman, G.: Possibility theory I, II, III. Int. J. General Systems 25 (1997), 291-323, 325-351, 353-371. DOI 10.1080/03081079708945160 | MR 1449009
[3] Faure, R., Heurgon, E.: Structures Ordonnées et Algèbres de Boole. Gauthier-Villars, Paris 1971. MR 0277440 | Zbl 0219.06001
[4] Fine, T. L.: Theories of Probability. An Examination of Foundations. Academic Press, New York - London 1973. MR 0433529 | Zbl 0275.60006
[5] Goguen, J. A.: ${\cal L}$-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174. DOI 10.1016/0022-247x(67)90189-8 | MR 0224391
[6] Halmos, P. R.: Measure Theory. D. van Nostrand, New York 1950. MR 0033869 | Zbl 0283.28001
[7] Kramosil, I.: Extensions of partial lattice-valued possibilistic measures from nested domains. Int. J. Uncertain. Fuzziness and Knowledge-Based Systems 14 (2006), 175-197. DOI 10.1142/s0218488506003935 | MR 2224969 | Zbl 1086.28009
[8] Kramosil, I., Daniel, M.: Statistical estimations of lattice-valued possibilistic distributions. In: Proc. Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU 2011 (W. Liu, ed.), LNCS (LNAI) 6717, Springer-Verlag Berlin - Heidelberg 2011, pp. 688-699. DOI 10.1007/978-3-642-22152-1_58 | MR 2831217
[9] Kramosil, I., Daniel, M.: Possibilistic distributions processed by probabilistic algorithms. Kybernetika, submitted for publication.
[10] Shannon, C. E.: The mathematical theory of communication. The Bell Systems Technical Journal 27 (1948), 379-423, 623-656. DOI 10.1002/j.1538-7305.1948.tb00917.x | MR 0026286 | Zbl 0126.35701
[11] Sikorski, R.: Boolean Algebras. Second edition. Springer, Berlin 1964. MR 0177920
[12] Zadeh, L. A.: Fuzzy sets. Inform. Control 8 (1965), 338-353. DOI 10.1016/s0019-9958(65)90241-x | MR 0219427 | Zbl 0942.00007
[13] Zadeh, L. A.: Probability measures of fuzzy events. J. Math. Anal. Appl. 23 (1968), 421-427. DOI 10.1016/0022-247x(68)90078-4 | MR 0230569 | Zbl 0174.49002
[14] Zadeh, L. A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1 (1978), 3-28. DOI 10.1016/0165-0114(78)90029-5 | MR 0480045 | Zbl 0377.04002

Partner of