×

zbMATH — the first resource for mathematics

The Goodman-Nguyen relation within imprecise probability theory. (English) Zbl 1433.60002
Summary: The Goodman-Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman-Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.

MSC:
60A05 Axioms; other general questions in probability
60A86 Fuzzy probability
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] de Finetti, B., Theory of probability, (1974), Wiley New York
[2] Goodman, I. R.; Nguyen, H. T., Conditional objects and the modeling of uncertainties, (Gupta, M.; Yamakawa, T., Fuzzy Computing, (1988), Elsevier (North-Holland) Amsterdam), 119-138
[3] Coletti, G.; Gilio, A.; Scozzafava, R., Comparative probability for conditional events: a new look through coherence, Theory Decis., 35, 237-258, (1993) · Zbl 0785.90006
[4] Coletti, G.; Scozzafava, R., Characterization of coherent conditional probabilities as a tool for their assessment and extension, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 4, 103-127, (1996) · Zbl 1232.03010
[5] Milne, P., Bruno de Finetti and the logic of conditional events, Br. J. Philos. Sci., 48, 195-232, (1997) · Zbl 0948.03015
[6] Pelessoni, R.; Vicig, P., The goodman-nguyen relation in uncertainty measurement, (Kruse, R.; Berthold, M. R.; Moewes, C.; Gil, M. A.; Grzegorzewski, P.; Hryniewicz, O., Synergies of Soft Computing and Statistics for Intelligent Data Analysis, (2013), Springer Berlin/Heidelberg), 37-44 · Zbl 1336.68252
[7] Pelessoni, R.; Vicig, P., Uncertainty modelling and conditioning with convex imprecise previsions, Int. J. Approx. Reason., 39, 297-319, (2005) · Zbl 1123.62077
[8] Pelessoni, R.; Vicig, P., Williams coherence and beyond, Int. J. Approx. Reason., 50, 612-626, (2009) · Zbl 1214.68403
[9] Walley, P., Statistical reasoning with imprecise probabilities, (1991), Chapman and Hall London · Zbl 0732.62004
[10] Weichselberger, K., Elementare grundbegriffe einer allgemeineren wahrscheinlichkeitsrechnung I: intervallwahrscheinlichkeit als umfassendes konzept, vol. 1, (2001), Physica Verlag Heidelberg · Zbl 0979.60001
[11] Williams, P. M., Notes on conditional previsions, Int. J. Approx. Reason., 44, 366-383, (2007) · Zbl 1114.60005
[12] Föllmer, H.; Schied, A., Convex measures of risk and trading constraints, Finance Stoch., 6, 429-447, (2002) · Zbl 1041.91039
[13] Holzer, S., On coherence and conditional prevision, Boll. Unione Mat. Ital. C (6), 4, 441-460, (1985) · Zbl 0584.60001
[14] Pelessoni, R.; Vicig, P., Bayes’ theorem bounds for convex lower previsions, J. Stat. Theory Pract., 3, 85-101, (2009) · Zbl 1211.62012
[15] de Finetti, B., La logique de la probabilité, (Actes du Congrès International de Philosophie Scientifique, vol. IV, (1936), Hermann Paris), Philos. Stud., 77, 181-190, (1995), English version: The logic of probability
[16] Milne, P., Bets and boundaries: assigning probabilities to imprecisely specified events, Stud. Log., 90, 425-453, (2008) · Zbl 1159.03309
[17] Gilio, A.; Sanfilippo, G., Quasi conjunction, quasi disjunction, t-norms and t-conorms: probabilistic aspects, Inf. Sci., 245, 146-167, (2013) · Zbl 1320.68188
[18] Goodman, I. R.; Nguyen, H. T.; Walker, E. A., Conditional inference and logic for intelligent systems: A theory of measure-free conditioning, (1991), Elsevier (North-Holland) Amsterdam
[19] Pelessoni, R.; Vicig, P.; Zaffalon, M., Inference and risk measurement with the pari-mutuel model, Int. J. Approx. Reason., 51, 1145-1158, (2010) · Zbl 1237.91136
[20] de Finetti, B., La prévision: ses lois logiques, ses sources subjectives, (Breakthroughs in Statistics, vol. I, (1992), Springer), 7, 134-174, (1937), English version: Foresight: its logical laws, its subjective sources · JFM 63.1070.02
[21] Baroni, P.; Pelessoni, R.; Vicig, P., Generalizing Dutch risk measures through imprecise previsions, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 17, 153-177, (2009) · Zbl 1162.91396
[22] Dolecki, S.; Greco, G. H., Niveloids, Topol. Methods Nonlinear Anal., 5, (1995) · Zbl 0839.49010
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.