Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We study infinite exchangeability assessments for them, and give a counterpart of de Finetti’s infinite representation theorem. We show how the infinite representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability.


desirability weak desirability sets of desirable gambles coherence exchangeability representation natural extension updating 


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  1. 1.
    De Cooman, G., Quaeghebeur, E.: Exchangeability for sets of desirable gambles. In: Augustin, T., Coolen, F.P.A., Moral, S., Troffaes, M.C.M. (eds.) ISIPTA 2009: proceedings of the Sixth International Symposium on Imprecise Probabilities: Theories and Applications, pp. 159–168. SIPTA, Durham (2009), Google Scholar
  2. 2.
    De Cooman, G., Quaeghebeur, E., Miranda, E.: Exchangeable lower previsions. Bernoulli 15(3), 721–735 (2009),, an expanded version can be found at MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de Finetti, B.: La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré 7, 1–68 (1937); English translation in [8]zbMATHGoogle Scholar
  4. 4.
    de Finetti, B.: Teoria delle Probabilità. Einaudi, Turin (1970)Google Scholar
  5. 5.
    de Finetti, B.: Theory of Probability: A Critical Introductory Treatment. John Wiley & Sons, Chichester (1974-1975), English translation of [4], two volumeszbMATHGoogle Scholar
  6. 6.
    Heath, D.C., Sudderth, W.D.: De Finetti’s theorem on exchangeable variables. The American Statistician 30, 188–189 (1976)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Johnson, N.L., Kotz, S., Balakrishnan, N.: Discrete Multivariate Distributions. Wiley Series in Probability and Statistics. John Wiley and Sons, New York (1997)zbMATHGoogle Scholar
  8. 8.
    Kyburg Jr., H.E., Smokler, H.E. (eds.): Studies in Subjective Probability. Wiley, New York (1964); 2nd edn. (with new material) (1980)zbMATHGoogle Scholar
  9. 9.
    Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Schechter, E.: Handbook of Analysis and Its Foundations. Academic Press, San Diego (1997)zbMATHGoogle Scholar
  11. 11.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)CrossRefzbMATHGoogle Scholar
  12. 12.
    Walley, P.: Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24, 125–148 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gert de Cooman
    • 1
  • Erik Quaeghebeur
    • 1
    • 2
  1. 1.SYSTeMS Research GroupGhent UniversityZwijnaardeBelgium
  2. 2.Department of PhilosophyCarnegie Mellon UniversityPittsburghUnited States

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