×

zbMATH — the first resource for mathematics

Finitely maxitive conditional possibilities, Bayesian-like inference, disintegrability and conglomerability. (English) Zbl 1383.62060
Summary: The aim of the paper is to study Bayesian-like inference processes involving coherent finitely maxitive \(T\)-conditional possibilities assessed on infinite sets of conditional events. Coherence of an assessment consisting of an arbitrary possibilistic prior and an arbitrary possibilistic likelihood function is proved, thus a closed form expression for the envelopes of the relevant joint and posterior possibilities is given when \(T\) is the minimum or a strict t-norm. The notions of disintegrability and conglomerability are also studied and their relevance in the infinite version of the possibilistic Bayes formula is highlighted.

MSC:
62F15 Bayesian inference
60A05 Axioms; other general questions in probability
62F86 Parametric inference and fuzziness
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Armstrong, T. E., Conglomerability of probability measures on Boolean algebras, J. Math. Anal. Appl., 150, 2, 335-358, (1990) · Zbl 0714.28004
[2] Baioletti, M.; Coletti, G.; Petturiti, D.; Vantaggi, B., Inferential models and relevant algorithms in a possibilistic framework, Int. J. Approx. Reason., 52, 5, 580-598, (2011) · Zbl 1214.68393
[3] Baioletti, M.; Petturiti, D., Algorithms for possibility assessments: coherence and extension, Fuzzy Sets Syst., 169, 1, 1-25, (2011) · Zbl 1214.68394
[4] Berti, P.; Rigo, P., Weak disintegrability as a form of preservation of coherence, J. Ital. Stat. Soc., 1, 2, 161-181, (1992) · Zbl 1446.60003
[5] Bouchon-Meunier, B.; Coletti, G.; Marsala, C., Conditional possibility and necessity, (Bouchon-Meunier, B.; Gutiérrez-Ríos, J.; Magdalena, L.; Yager, R. R., Technologies for Constructing Intelligent Systems 2, Studies in Fuzziness and Soft Computing, vol. 90, (2002), Physica-Verlag HD), 59-71 · Zbl 1015.68191
[6] Bouchon-Meunier, B.; Coletti, G.; Marsala, C., Independence and possibilistic conditioning, Ann. Math. Artif. Intell., 35, 1-4, 107-123, (2002) · Zbl 1004.60001
[7] Cifarelli, D. M.; Regazzini, E., De Finetti’s contribution to probability and statistics, Stat. Sci., 11, 4, 253-282, (1996) · Zbl 0955.01552
[8] Coletti, G.; Petturiti, D.; Vantaggi, B., Bayesian inference: the role of coherence to deal with a prior belief function, Stat. Methods Appl., 23, 4, 519-545, (2014)
[9] Coletti, G.; Petturiti, D.; Vantaggi, B., Coherent T-conditional possibility envelopes and nonmonotonic reasoning, (Laurent, A.; Strauss, O.; Bouchon-Meunier, B.; Yager, R. R., Information Processing and Management of Uncertainty in Knowledge-Based Systems, Communications in Computer and Information Science, vol. 444, (2014), Springer International Publishing), 446-455
[10] Coletti, G.; Petturiti, D.; Vantaggi, B., Possibilistic and probabilistic likelihood functions and their extensions: common features and specific characteristics, Fuzzy Sets Syst., 250, 25-51, (2014) · Zbl 1334.60004
[11] Coletti, G.; Scozzafava, R., From conditional events to conditional measures: a new axiomatic approach, Ann. Math. Artif. Intell., 32, 1-4, 373-392, (2001) · Zbl 1314.68306
[12] Coletti, G.; Scozzafava, R., Probabilistic logic in a coherent setting, Trends in Logic, vol. 15, (2002), Kluwer Academic Publisher Dordrecht/Boston/London · Zbl 1005.60007
[13] Coletti, G.; Vantaggi, B., Possibility theory: conditional independence, Fuzzy Sets Syst., 157, 11, 1491-1513, (2006) · Zbl 1092.68094
[14] Coletti, G.; Vantaggi, B., T-conditional possibilities: coherence and inference, Fuzzy Sets Syst., 160, 3, 306-324, (2009) · Zbl 1178.60006
[15] De Baets, B.; Tsiporkova, E.; Mesiar, R., Conditioning in possibility theory with strict order norms, Fuzzy Sets Syst., 106, 2, 221-229, (1999) · Zbl 0985.28015
[16] de Cooman, G., Possibility theory I: the measure- and integral-theoretic groundwork, Int. J. Gen. Syst., 25, 291-323, (1997) · Zbl 0955.28012
[17] de Cooman, G., Possibility theory II: conditional possibility, Int. J. Gen. Syst., 25, 4, 325-351, (1997) · Zbl 0955.28013
[18] de Cooman, G., Integration and conditioning in numerical possibility theory, Ann. Math. Artif. Intell., 32, 1-4, 87-123, (2001) · Zbl 1314.28012
[19] de Cooman, G.; Kerre, E. E., Possibility and necessity integrals, Fuzzy Sets Syst., 77, 2, 207-227, (1996) · Zbl 0872.28010
[20] de Finetti, B., Sulla proprietà conglomerativa delle probabilità subordinate, Atti R. Accad. Naz. Lincei, Ser. VI, Rend., 12, 278-282, (1930) · JFM 56.0445.02
[21] de Finetti, B.; de Finetti, B., Aggiunta alla nota sull’assiomatica Della probabilità, Ann. Triest., Ann. Triest., 20, 3-20, (1949) · Zbl 0044.13402
[22] de Finetti, B., Probability, induction and statistics: the art of guessing, (1972), John Wiley & Sons London, New York, Sydney, Toronto · Zbl 0275.60001
[23] Dempster, A. P., Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Stat., 38, 2, 325-339, (1967) · Zbl 0168.17501
[24] Dubins, L. E., Finitely additive conditional probabilities, conglomerability and disintegrations, Ann. Probab., 3, 1, 89-99, (1975) · Zbl 0302.60002
[25] Dubois, D.; Prade, H., Possibility theory: an approach to computerized processing of uncertainty, (1988), Plenum Press New York and London
[26] El Rayes, A. B.; Morsi, N. N., Generalized possibility measures, Inf. Sci., 79, 3-4, 201-222, (1994) · Zbl 0812.28012
[27] Hisdal, E., Conditional possibilities independence and noninteraction, Fuzzy Sets Syst., 1, 4, 283-297, (1978) · Zbl 0393.94050
[28] Klement, E. P.; Mesiar, R.; Pap, E., Triangular norms, vol. 8, (2000), Kluwer Academic Publisher Dordrecht/Boston/London
[29] Miranda, E.; Zaffalon, M.; de Cooman, G., Conglomerable natural extension, Int. J. Approx. Reason., 53, 8, 1200-1227, (2012) · Zbl 1287.60006
[30] Petturiti, D., Coherent conditional possibility theory and possibilistic graphical modeling in a coherent setting, (2013), Università degli Studi di Perugia Perugia, Italy, PhD thesis
[31] Regazzini, E., De Finetti’s coherence and statistical inference, Ann. Stat., 15, 2, 845-864, (1987) · Zbl 0653.62003
[32] Schervish, M. J.; Seidenfeld, T.; Kadane, J. B., The extent of non-conglomerability of finitely additive probabilities, Z. Wahrscheinlichkeitstheor. Verw. Geb., 66, 205-226, (1984) · Zbl 0525.60003
[33] Scozzafava, R., Probabilità σ-additive e non, Boll. UMI, 1-A, 6, 1-3, (1982) · Zbl 0484.60003
[34] Seidenfeld, T.; Schervish, M. J.; Kadane, J. B., Non-conglomerability for finite-valued, finitely additive probability, Special Issue on Bayesian Analysis, Indian J. Stat., A, 60, 476-491, (1998) · Zbl 0978.60005
[35] Shilkret, N., Maxitive measure and integration, Indag. Math. (Proc.), 74, 109-116, (1971) · Zbl 0218.28005
[36] Sugeno, M., Theory of fuzzy integrals and its applications, (1974), Tokyo Institute of Technology Tokyo, Japan, PhD thesis
[37] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall London · Zbl 0732.62004
[38] Walley, P.; de Cooman, G., Coherence of rules for defining conditional possibility, Int. J. Approx. Reason., 21, 1, 63-107, (1999) · Zbl 0957.68115
[39] Weber, S., ⊥-decomposable measures and integrals for Archimedean t-conorms ⊥, J. Math. Anal. Appl., 101, 1, 114-138, (1984) · Zbl 0614.28019
[40] Weber, S., Two integrals and some modified versions - critical remarks, Fuzzy Sets Syst., 20, 1, 97-105, (1986) · Zbl 0595.28012
[41] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst., 1, 1, 3-28, (1978) · Zbl 0377.04002
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.