×

zbMATH — the first resource for mathematics

Do inferential processes affect uncertainty frameworks? (English) Zbl 1315.68235
Summary: This paper studies the connections among different (comparative or numerical) degrees of belief. In particular we consider, in turn, a comparative probability or possibility on a given Boolean algebra and we prove that their upper extensions to a different Boolean algebra are, respectively, a comparative plausibility or possibility. On the other hand, in general the upper extension of a comparative necessity is simply a comparative capacity. Moreover, by considering a suitable condition of weak logical independence between the two Boolean algebras, we prove that the upper ordinal relation is a comparative possibility in all the aforementioned cases. We consider specifically also the lower ordinal relations, since they may not be the comparative dual relation of the upper ones.

MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baioletti, M.; Petturiti, D., Algorithms for possibility assessmentscoherence and extension, Fuzzy Set Syst., 169, 1-25, (2011) · Zbl 1214.68394
[2] Baioletti, M.; Petturiti, D.; Coletti, G.; Vantaggi, B., Inferential models and relevant algorithms in a possibilistic framework, Int. J. Approx. Reason., 52, 580-598, (2011) · Zbl 1214.68393
[3] Baroni, P.; Vicig, P., An uncertainty interchange format with imprecise probabilities, Int. J. Approx. Reason., 40, 3, 147-180, (2005) · Zbl 1110.68145
[4] Capotorti, A.; Coletti, G.; Vantaggi, B., Non-additive ordinal relations representable by lower and upper probabilities, Kybernetika, 34, 1, 79-90, (1998) · Zbl 1274.68518
[5] Coletti, G., Coherent qualitative probability, J. Math. Psychol., 34, 297-310, (1990) · Zbl 0713.60003
[6] Coletti, G.; Scozzafava, R., From conditional events to conditional measuresa new axiomatic approach, Ann. Math. Artif. Intell., 32, 373-392, (2001) · Zbl 1314.68306
[7] Coletti, G.; Scozzafava, R., Probabilistic logic in a coherent setting, (2002), Kluwer Dordrecht · Zbl 1005.60007
[8] Coletti, G.; Scozzafava, R., Toward a general theory of conditional beliefs, Int. J. Intell. Syst., 21, 229-259, (2006) · Zbl 1160.68582
[9] G. Coletti, R. Scozzafava, B. Vantaggi, Possibility measures through a probabilistic inferential process, in: Proceedings of the NAFIPS 2008, IEEE CN:CFP08750-CDR Omnipress, 2008. · Zbl 1320.68181
[10] G. Coletti, R. Scozzafava, B. Vantaggi, Inferential processes leading to possibility and necessity, Inf. Sci., in press, http://dx.doi.org/10.1016/j.ins.2012.10.034 (online since 6/11/2012). · Zbl 1320.68181
[11] Coletti, G.; Scozzafava, R.; Vantaggi, B., A bridge between probability and possibility in a comparative framework, Lecture Notes in Computer Science, 6717, 557-568, (2011) · Zbl 1341.68242
[12] Coletti, G.; Vantaggi, B., Representability of ordinal relations on a set of conditional events, Theory Decision, 60, 137-174, (2006) · Zbl 1119.91029
[13] Coletti, G.; Vantaggi, B., A view on conditional measures through local representability of binary relations, Int. J. Approx. Reason., 40, 268-283, (2008) · Zbl 1184.68500
[14] Coletti, G.; Vantaggi, B., T-conditional possibilitiescoherence and inference, Fuzzy Set Syst., 160, 306-324, (2009) · Zbl 1178.60006
[15] de Finetti, B., Sul significato soggettivo Della probabilit√†, Fundam. Mat., 17, 293-329, (1931) · JFM 57.0608.07
[16] Delgado, M.; Moral, S., On the concept of possibility-probability consistency, Fuzzy Set Syst., 21, 311-318, (1987) · Zbl 0618.60003
[17] Dempster, A. P., Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Stat., 38, 325-339, (1967) · Zbl 0168.17501
[18] Dennemberg, D., Non-additive measure and integral, (1997), Kluwer Berlin
[19] Dubois, D., Belief structure, possibility theory and decomposable confidence measures on finite sets, Comput. Artif. Intell., 5, 403-416, (1986) · Zbl 0657.60006
[20] D. Dubois, Fuzzy measures on finite scales as families of possibility measures, in: Proceedings of the European Society for Fuzzy Logic and Technology (EUSFLAT-LFA), Aix-Les-Bains, France, 2011. · Zbl 1254.28017
[21] Dubois, D.; Nguyen, H. T.; Prade, H., Possibility theory, probability and fuzzy setsmisunderstandings, bridges and gaps, (Dubois, D.; Prade, H., Fundamentals of Fuzzy Sets, (2000), Kluwer Boston) · Zbl 0978.94052
[22] D. Dubois, H. Prade, Upper and lower possibilities induced by a multivalued mapping, in: Proceedings of the IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, 1983, pp. 152-174. · Zbl 0562.94023
[23] Dubois, D.; Prade, H., When upper probabilities are possibility measures, Fuzzy Set Syst., 49, 65-74, (1992) · Zbl 0754.60003
[24] Dubois, D.; Prade, H., Evidence measures based on fuzzy information, Automatica, 21, 547-562, (1985) · Zbl 0596.62007
[25] Dubois, D.; Prade, H., Qualitative possibility theory and its probabilistic connections, (Grzegorzewski, P.; etal., Soft Methods in Probability, Statistics and Data Analysis, (2002), Physica Verlag Heidelberg)
[26] Dubois, D.; Prade, H.; Smets, P., A definition of subjective possibility, Oper. Res. Decisions, 4, 7-22, (2003)
[27] Giles, R., Foundations for a theory of possibility, (Gupta, M. M.; Sanchez, M. M.E., Fuzzy Information and Decision Processes, (1982), North-Holland Amsterdam) · Zbl 0514.94028
[28] Gilio, A., Penalization criterion and coherence conditions in subjective assessment of probabilities, Boll. Unione Mat. Ital., 4B, 645-660, (1990) · Zbl 0716.60002
[29] Halpern, J., Reasoning about uncertainty, (2003), The MIT Press Boston · Zbl 1090.68105
[30] Koopman, B. O., The axioms and algebra of intuitive probability, Ann. Math., 41, 269-292, (1940) · JFM 66.1195.01
[31] Kraft, C. H.; Pratt, J. W.; Seidenberg, A., Intuitive probability on finite sets, Ann. Math. Stat., 30, 408-419, (1959) · Zbl 0173.19606
[32] H. Prade, A. Rico, Possibilistic Evidence, in: Lecture Notes, Lecture Notes in Artificial Intelligence, vol. 6717, 2011, pp. 713-724. · Zbl 1341.68263
[33] Scott, D., Measurement structures and linear inequalities, J. Math. Psychol., 1, 233-247, (1964) · Zbl 0129.12102
[34] Sudkamp, T., On probability-possibility transformations, Fuzzy Set Syst., 51, 311-318, (1992)
[35] Tsiporkova, E.; De Baets, B., A general framework for upper and lower possibilities and necessities, Int. J. Uncertainty Fuzziness Knowledge-Based Syst., 6, 1-34, (1998) · Zbl 1065.03510
[36] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall London · Zbl 0732.62004
[37] Wong, S. K.M.; Yao, Y. Y.; Bollmann, P.; Burger, H. C., Axiomatization of qualitative belief structure, IEEE Trans. Syst. Man Cybern., 21, 726-734, (1991) · Zbl 0737.60006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.