Notes on desirability and conditional lower previsions

Article

Abstract

We detail the relationship between sets of desirable gambles and conditional lower previsions. The former is one the most general models of uncertainty. The latter corresponds to Walley’s celebrated theory of imprecise probability. We consider two avenues: when a collection of conditional lower previsions is derived from a set of desirable gambles, and its converse. In either case, we relate the properties of the derived model with those of the originating one. Our results constitute basic tools to move from one formalism to the other, and thus to take advantage of work done in the two fronts.

Keywords

Lower and upper previsions Sets of desirable gambles Coherence Natural extension Equal expressivity 

Mathematics Subject Classifications (2010)

28E05 03B48 28A12 60A86 

References

  1. 1.
    Couso, I., Moral, S.: Sets of desirable gambles and credal sets. In: Augustin, T., Coolen, F., Moral, S., Troffaes, M.C.M. (eds.) ISIPTA ’09—Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications, pp. 99–108. Manno, Switzerland. SIPTA (2009)Google Scholar
  2. 2.
    de Campos, L.M., Lamata, M.T., Moral, S.: The concept of conditional fuzzy measures. Int. J. Intell. Syst. 5, 237–246 (1990)MATHCrossRefGoogle Scholar
  3. 3.
    De Cooman, G., Hermans, F.: Coherent immediate prediction: bridging two theories of imprecise probability. Artif. Intell. 172(11), 1400–1427 (2008)MATHCrossRefGoogle Scholar
  4. 4.
    de Cooman, G., Miranda, E.: Symmetry of models versus models of symmetry. In: Harper, W.L., Wheeler, G.R. (eds.) Probability and Inference: Essays in Honor of Henry E. Kyburg, Jr., pp. 67–149. King’s College Publications (2007)Google Scholar
  5. 5.
    De Cooman, G., Quaeghebeur, E.: Exchangeability and sets of desirable gambles. Int. J. Approx. Reason. (2011, in press)Google Scholar
  6. 6.
    de Cooman, G., Zaffalon, M.: Updating beliefs with incomplete observations. Artif. Intell. 159(1–2), 75–125 (2004)MATHGoogle Scholar
  7. 7.
    Fagin, R., Halpern, J.Y.: A new approach to updating beliefs. In: Bonissone, P.P., Henrion, M., Kanal, L.N., Lemmer, J.F. (eds.) Uncertainty in Artificial Intelligence, vol. 6, pp. 347–374. North-Holland, Amsterdam (1991)Google Scholar
  8. 8.
    Jaffray, J.-Y.: Bayesian updating and belief functions. IEEE Trans. Syst. Man Cybern. 22, 1144–1152 (1992)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kikuti, D., Cozman, F.G., Shirota Filhoa, R.: Sequential decision making with partially ordered preferences. Artif. Intell. 175(7–8), 1346–1365 (2011)CrossRefGoogle Scholar
  10. 10.
    Levi, I.: Potential surprise: its role in inference and decision making. In: Cohen, L.J., Hesse, M. (eds.) Applications of Inductive Logic, pp. 1–27. Clarendon Press, Oxford (1980)Google Scholar
  11. 11.
    Miranda, E.: A survey of the theory of coherent lower previsions. Int. J. Approx. Reason. 48(2), 628–658 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Miranda, E.: Updating coherent previsions on finite spaces. Fuzzy Sets Syst. 160(9), 1286–1307 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Miranda, E., de Cooman, G.: Coherence and Independence in Non-Linear Spaces. Technical Report. Downloadable at http://bellman.ciencias.uniovi.es/~emiranda/wp05-10.pdf (2005)
  14. 14.
    Miranda, E., Zaffalon, M.: Conditional models: coherence and inference through sequences of joint mass functions. J. Stat. Plan. Inference 140(7), 1805–1833 (2010)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Moral, S.: Epistemic irrelevance on sets of desirable gambles. Ann. Math. Artif. Intell. 45, 197–214 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pelessoni, R., Vicig, P.: Williams coherence and beyond. Int. J. Approx. Reason. 50(4), 612–626 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Smith, C.A.B.: Consistency in statistical inference and decision. J. R. Stat. Soc., Ser. A 23, 1–37 (1961)MATHGoogle Scholar
  18. 18.
    Vicig, P., Zaffalon, M., Cozman, F.: Notes on ‘Notes on conditional previsions’. Int. J. Approx. Reason. 44(3), 358–365 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)MATHGoogle Scholar
  20. 20.
    Walley, P.: Inferences from multinomial data: learning about a bag of marbles. J. R. Stat. Soc., Ser. B 58, 3–57 (1996) With discussionMathSciNetMATHGoogle Scholar
  21. 21.
    Walley, P.: Towards a unified theory of imprecise probability. Int. J. Approx. Reason. 24(2–3), 125–148 (2000)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Walley, P., Pelessoni, R., Vicig, P.: Direct algorithms for checking consistecy and making inferences for conditional probability assessments. J. Stat. Plan. Inference 126(1), 119–151 (2004)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Williams, P.M.: Coherence, strict coherence and zero probabilities. In: Proceedings of the Fifth International Congress in Logic, Methodology, and Philosophy of Science, pp. 29–33 (1975)Google Scholar
  24. 24.
    Williams, P.M.: Notes on Conditional Previsions. Technical report, School of Mathematical and Physical Science, University of Sussex, UK (1975). Reprinted in International Journal of Approximate Reasoning, vol. 44, pp. 366–383, 2007Google Scholar
  25. 25.
    Williams, P.M.: Notes on conditional previsions. Int. J. Approx. Reason. 44(3), 366–383 (2007)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversity of OviedoOviedoSpain
  2. 2.IDSIAMannoSwitzerland

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