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Coherent Upper and Lower Conditional Previsions Defined by Hausdorff Outer and Inner Measures

Chapter
Part of the Mathematical Engineering book series (MATHENGIN, volume 3)

Abstract

A new model of coherent upper conditional previsions is proposed to represent uncertainty and to make previsions in complex systems. It is defined by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero or infinity in its Hausdorff dimension, it is defined by a 0-1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension, it is proven that a coherent upper conditional prevision is uniquely represented by the Choquet integral with respect to the upper conditional probability defined by Hausdorff outer measure if and only if it is monotone, comonotonically additive, submodular and continuous from below.Moreover sufficient conditions are given such that the upper conditional previsions satisfy the disintegration property and the conglomerability principle.

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References

  1. 1.
    Artzner, P., Delbaen, F., Eber, J., Heath, D.: Coherent measures of risk. Math. Finance 3, 203–228 (1999)MathSciNetGoogle Scholar
  2. 2.
    Bassanezi, R., Greco, G.: Sull’additivita’ dell’ integrale. Rend. Sem. Mat. Univ. Padova 72, 249–275 (1984). (In Italian)Google Scholar
  3. 3.
    Billingsley, P.: Probability and measure. Chapman and Hall (1991)Google Scholar
  4. 4.
    Cabrelli, C., Hare, K.E., Molter, U.M.: Classifying Cantor sets by their fractal dimension. Proceedings of the American Mathematical Society 139(11), 3965–3974 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cabrelli, C., Mendivil, F., Molter, U.M., Shonkwiler, R.: On the Hausdorff h-measure of Cantor sets. Pacific Journal of Mathematics 217(1), 45–60 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    de Cooman, G., Troffaes, M., Miranda, E.: n-monotone exact functionals. Journal of Mathematical Analysis and Applications 347, 133–146 (2008)Google Scholar
  7. 7.
    Croydon, D.: Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. Annals of Probability 37(3), 946–978 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Denneberg, D.: Non-additive measure and integral. Kluwer Academic Publishers (1994)Google Scholar
  9. 9.
    Doria, S.: Probabilistic independence with respect to upper and lower conditional probabilities assigned by Hausdorff outer and inner measures. International Journal of Approximate Reasoning 46, 617–635 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Doria, S.: Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdorff outer measure. Submitted to Annals of Operations Research (2010)Google Scholar
  11. 11.
    Doria, S.: Coherent upper conditional previsions and their integral representation with respect to Hausdorff outer measures. Advances in Intelligent and Soft Computing pp. 209–216 (2010)Google Scholar
  12. 12.
    Doria, S.: Stochastic independence with respect to upper and lower conditional probabilities assigned by Hausdorff outer and inner measures. Stochastic Control 46, 87–101 (2010)Google Scholar
  13. 13.
    Dubins, L.: Finitely additive conditional probabilities, conglomerability and disintegrations. Annals of Probability 3, 89–99 (1975)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Falconer, K.: The geometry of fractals sets. Cambridge University Press (1986)Google Scholar
  15. 15.
    de Finetti, B.: Teoria della Probabilita’. Einaudi (1970). (In Italian)Google Scholar
  16. 16.
    de Finetti, B.: Induction and Statistics. Wiley (1972)Google Scholar
  17. 17.
    Greco, G.: Sur la mesurabilit´e d’une fonction num´erique par rapport `a une famille d’ensembles. Rend. Sem. Mat. Univ. Padova 65, 21–42 (1981). (In French)Google Scholar
  18. 18.
    Greco, G.: Sulla rappresentazione di funzionali mediante integrali. Rend. Sem. Mat. Univ. Padova pp. 21–42 (1982). (In Italian)Google Scholar
  19. 19.
    Kadane, J.B., Schervish, M., Seidenfeld, T.: Statistical implications of finitely additive probability. Bayesian Inference and Decision Techniques With Applications pp. 59–76 (1986)Google Scholar
  20. 20.
    Koch, G.: La matematica del probabile. Aracne Editrice (1997). (In Italian)Google Scholar
  21. 21.
    Maaß, S.: Exact functionals and their core. Statistical papers 45(1), 75–93 (2002)Google Scholar
  22. 22.
    Miranda, E., Zaffalon, M.: Conditional models: coherence and inference trough sequences of joint mass functions. Journal of Statistical Planning and Inference 140(7), 1805–1833 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Narukawa, Y., Murofushi, T., Sugeno, M.: Regular fuzzy measure and representation of comonotonically additive functionals. Fuzzy Sets and Systems 112, 177–186 (2000)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Perkins, E.: The exact Hausdorff measure of the level sets of Brownian motion. Probability theory and related fields (1981)Google Scholar
  25. 25.
    Regazzini, E.: Finitely additive conditional probabilities. Rend. Sem. Mat. Fis. 58(3), 373–388 (1985)Google Scholar
  26. 26.
    Regazzini, E.: De Finetti’s coherence and statistical inference. The Annals of Statistics 15(2), 845–864 (1987)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Rogers, C.: Hausdorff measures. Cambridge University Press (1970)Google Scholar
  28. 28.
    Schervish, M., Seidenfeld, T., Kadane, J.: The extent of non-conglomerability of finitely additive probabilities. Z. Warsch.Verw.Gebiete 66, 205–226 (1984)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Schmeidler, D.: Cores of exact games. I.J.Math.Anal.Appl. 40, 214–225 (1972)Google Scholar
  30. 30.
    Schmeidler, D.: Integral representation without additivity. Proceedindgs of the American Mathematical Society 97, 225–261 (1986)MathSciNetGoogle Scholar
  31. 31.
    Scozzafava, R.: Probabilita’ -additive e non. Bollettino U.M.I. 57(1-A), 1–33 (1986). (In Italian)Google Scholar
  32. 32.
    Vicig, P., Zaffalon, M., Cozman, F.: Notes on “Notes on conditional previsions”. International Journal of Approximate Reasoning 44, 358–365 (2007)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Walley, P.: Coherent lower (and upper) probabilities. Statistics Research Report, University of Warwick (1981)Google Scholar
  34. 34.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall (1991)Google Scholar
  35. 35.
    Williams, P.: Notes on conditional previsions. International Journal of Approximate Reasoning 44, 366–383 (2007)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of SciencesUniversity G. D’AnnunzioChieti-PescaraItaly

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