Kybernetika 50 no. 2, 189-215, 2014

Standard and nonstandard representability of positive uncertainty orderings

Andrea Capotorti, Giulianella Coletti and Barbara VantaggiDOI: 10.14736/kyb-2014-2-0189

Abstract:

Axioms are given for positive comparative probabilities and plausibilities defined either on Boolean algebras or on arbitrary sets of events. These axioms allow to characterize binary relations representable by either standard or nonstandard measures (i. e. taking values either on the real field or on a hyperreal field). We also study relations between conditional events induced by preferences on conditional acts.

Keywords:

comparative probability, comparative plausibilities, hyperreal field, representability by nonstandard measures

Classification:

60A05, 62C10, 91B08

paper.pdf

References:

  1. B. Ben Yaghlane, P. Smets and K. Mellouli: About conditional belief function independence. Lect. Notes in Comput. Sci. 2143 (2001), 340-349.   CrossRef
  2. S. Bernardi and G. Coletti: A Rational conditional utility model in a coherent framework. Lect. Notes in Comput. Sci. 2143 (2001), 108-119.   CrossRef
  3. L. Blume, A. Brandenburger and E. Dekel: Lexicographic probabilities and choice under uncertainty. Econometrica 59 (1991), 1, 61-79.   CrossRef
  4. A. Capotorti, G. Coletti and B. Vantaggi: Non-additive ordinal relations representable by lower or upper probabilities. Kybernetika 34 (1998), 10, 79-90.   CrossRef
  5. A. Capotorti, G. Coletti and B. Vantaggi: Preferences representable by a lower expectation: some characterizations. Theory and Decision 64 (2008), 119-146.   CrossRef
  6. A. Chateauneuf and J. Y. Jaffray: Archimedean qualitative probabilities. J. Math. Psychol. 28 (1984), 191-204.   CrossRef
  7. G. Coletti: Coherent qualitative probability. J. Math. Psychol. 34 (1990), 297--310.   CrossRef
  8. G. Coletti: Coherent numerical and ordinal probabilistic assessments. IEEE Tras. Systems, Man, and Cybernetics 24 (1994), 12, 1747--1754.   CrossRef
  9. G. Coletti and M. Mastroleo: Conditional belief functions: a comparison among different definitions. In: Proc. 7th Workshop on Uncertainty Processing (WUPES), 2006.   CrossRef
  10. G. Coletti and R. Scozzafava: Toward a general theory of conditional beliefs. Internat. J. of Intelligent Systems 21 (2006), 229-259.   CrossRef
  11. G. Coletti, R. Scozzafava and B. Vantaggi: Integrated likelihood in a finitely additive setting. Lect. Notes in Computer Science LNAI 5590 (2009), 554-565.   CrossRef
  12. G. Coletti and B. Vantaggi: Representability of ordinal relations on a set of conditional events. Theory and Decision 60 (2006), 137-174.   CrossRef
  13. G. Coletti and B. Vantaggi: A view on conditional measures through local representability of binary relations. Internat. J. Approximate Reasoning 60 (2006), 137-174.   CrossRef
  14. G. Coletti and B. Vantaggi: Conditional not-additive measures and fuzzy sets. In: Proc. ISIPTA 2013, pp. 67-76.   CrossRef
  15. B. de Finetti: Sul significato soggettivo delle probabilità. Fundam. Mat. 17 (1931), 293-329.   CrossRef
  16. B. de Finetti: La prevision: Ses lois logiques, ses sources subjectives. Ann. Inst. Henri Poincaré, Section B 7 (1937), l-68.   CrossRef
  17. B. de Finetti: Teoria della Probabilità. Einaudi, Torino 1970 (Engl. transl.) Theory of Probability, Wiley and Sons, London 1974.   CrossRef
  18. A. P. Dempster: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38 (1967), 325-339.   CrossRef
  19. A. P. Dempster: A generalization of Bayesian inference. The Royal Stat. Soc. B 50 (1968), 205-247.   CrossRef
  20. T. Denoeux and P. Smets: classification using belief functions: The relationship between the case-based and model-based approaches. IEEE Trans. on Systems, Man and Cybernetics B 36 (2006), 6, 1395-1406.   CrossRef
  21. L. E. Dubins: Finitely additive conditional probabilities, conglomerability and disintegration. Ann. Probab. 3 (1975), 89-99.   CrossRef
  22. D. Dubois, H. Fargier and B. Vantaggi: An axiomatization of conditional possibilistic preference functionals. Lect. Notes LNAI 4724 (2007), 803-815.   CrossRef
  23. W. Fenchel: Convex Cones Sets and Functions. Lectures at Princeton University, Princeton 1951.   CrossRef
  24. R. Fagin, J. Y. Halpern and N. Megido: A logic for reasoning about probabilities. Information and Computation 87 (1990), 78-128.   CrossRef
  25. J .Y. Halpern: Lexicographic probability, conditional probability, and nonstandard probability. Games and Economic Behavior 68 (2010), 1, 155-179.   CrossRef
  26. P. Ghirardato: Revisiting savage in a conditional world. Economic Theory 20 (2002), 83-92.   CrossRef
  27. S. Holzer: On coherence and conditional prevision. Bollettino UMI, Serie VI-C IV (1985), 1, 441-460.   CrossRef
  28. J. Y. Jaffray: Bayesian updating and belief functions. IEEE Trans. on Systems, Man, and Cybernetics 22 (1992), 1144-1152.   CrossRef
  29. B. O. Koopman: The axioms and algebra of intuitive probability. Ann. Math. 41 (1940), 269-292.   CrossRef
  30. C. Kraft, J. Pratt and A. Seidenberg: Intuitive probability on finite sets. Ann. Math. Statist. 30 (1959), 408-419.   CrossRef
  31. P. H. Krauss: Representation of conditional probability measures on Boolean algebras. Acta Mathematica Academiae Sceintiarum Hungaricae 19 (1068), 3-4, 229-241.   CrossRef
  32. D. Lehmann: Generalized qualitative probability: Savage revisited. In: Proc. UAI'96, pp. 381-388.   CrossRef
  33. L. Narens: Minimal conditions for additive conjoint measurement and qualitative probability. J. Math. Psychol. 11 (1974), 404-430.   CrossRef
  34. J. Paris: A note on the Dutch Book method. In: Proc. Second International Symposium on Imprecise Probabilities and their Applications (G. De Cooman, T. Fine, and T. Seidenfeld, eds.), ISIPTA 2001, Shaker Publishing Company, Ithaca, pp. 301-306.   CrossRef
  35. A. Rényi: On conditional probability spaces generated by a dimensionally ordered set of measures. Theor. Probab. Appl. 1 (1956), 61-71.   CrossRef
  36. E. Regazzini: Finitely additive conditional probabilities. Rendiconti Sem. Mat. Fis. Milano 55 (1985), 69-89.   CrossRef
  37. G. Regoli: Rational comparisons and numerical representation. In: Decision Theory and Decision Analysis: Trends and Challenges, Academic Press, New York 1994.   CrossRef
  38. A. Robinson: Non-Standard Analysis. North Holland, Amsterdam 1966.   CrossRef
  39. L. J. Savage: The Foundations of Statistics. Wiley, New York 1954.   CrossRef
  40. G. Shafer: Allocations of probability. Ann. Probab. 7 (1979), 827-839.   CrossRef
  41. B. Vantaggi: Incomplete preferences on conditional random quantities: representability by conditional previsions. Math. Soc. Sci. 60 (2010), 104-112.   CrossRef
  42. P. Walley: Belief function representations of statistical evidence. Ann. Statist. 4 (1987), 1439-1465.   CrossRef
  43. P. Walley: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London 1991.   CrossRef
  44. P. M. Williams: Notes on Conditional Previsions. Working Paper School of Mathematical and Physical Sciences, The University of Sussex, 1975.   CrossRef
  45. S. K. M. Wong, Y. Y. Tao, P. Bollmann and H. C. Burger: Axiomatization of qualitative belief structure. IEEE Trans. Systems, Man, and Cybernet. 21 (1991), 726-734.   CrossRef