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Preference orderings represented by coherent upper and lower conditional previsions

  • Serena DoriaEmail author
Article
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Abstract

Preference orderings assigned by coherent lower and upper conditional previsions are defined and they are considered to define maximal random variables and Bayes random variables. Sufficient conditions are given such that a random variable is maximal if and only if it is a Bayes random variable. In a metric space preference orderings represented by coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures are given.

Keywords

Preference ordering Coherent upper and lower conditional previsions Choquet integral Disintegration property Hausdorff outer measures 

Notes

Acknowledgements

The author is grateful to two anonymous referees for their comments.

References

  1. Anscombe, F. J., & Aumann, R. J. (1963). A definition of subjective probability. The Annals of Statistics, 34(1), 199–205.CrossRefGoogle Scholar
  2. Billingsley, P. (1986). Probability and measure. Oxford: Wiley.Google Scholar
  3. Choquet, G. (1953). Theory of capacities. Annales de l’institut Fourier, 5, 131–295.CrossRefGoogle Scholar
  4. de Finetti, B. (1972). Probability, induction and statistics. New York: Wiley.Google Scholar
  5. de Finetti, B. (1974). Theory of probability. London: Wiley.Google Scholar
  6. Denneberg, D. (1994). Non-additive measure and integral. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
  7. Doria, S. (2007). 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.CrossRefGoogle Scholar
  8. Doria, S. (2008). Convergences of random variables with respect to coherent upper probabilities defined by Hausdorff outer measures, Advances in Soft Computing (Vol. 48, pp. 281–288). New York: Springer.Google Scholar
  9. Doria, S. (2010). Coherent upper conditional previsions and their integral representation with respect to Hausdorff outer measures Combining Soft Computing and Stats Methods, C.Borgelt et al. (Eds.) 209-216.Google Scholar
  10. Doria, S. (2011). Coherent upper and lower conditional previsions defined by hausdorff outer and inner measures. Modeling, Design, and Simulation Systems with Uncertainties, Mathematical Engineering Springer, 3, 175–195.CrossRefGoogle Scholar
  11. Doria, S. (2012). Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdorff outer measure. Annals of Operations Research, 195, 33–48.CrossRefGoogle Scholar
  12. Doria, S. (2014). Symmetric coherent upper conditional prevision by the Choquet integral with respect to Hausdorff outer measure. Annals of Operations Research, 229, 377–396.CrossRefGoogle Scholar
  13. Doria, S. (2015). Coherent conditional measures of risk defined by the Choquet integral with respect to Hausdorff outer measure and stochastic independence in risk management. International Journal of approximate Reasoning, 65, 1–10.CrossRefGoogle Scholar
  14. Doria, S. (2017). On the disintegration property of a coherent upper conditional prevision by the Choquet integral with respect to its associated Hausdorff outer measure. Annals of Operations Research, 256(2), 253–269.CrossRefGoogle Scholar
  15. Di Cencio, A., & Doria, S. (2017). Probabilistic Analysis of Suture Lines Developed in Ammonites: The Jurassic Examples of Hildocerataceae and Hammatocerataceae. Mathematical Geosciences, 49(6), 737–750.CrossRefGoogle Scholar
  16. Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. The Annals of Probability, 3, 89–99.CrossRefGoogle Scholar
  17. Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of economics, 75, 643–669.CrossRefGoogle Scholar
  18. Falconer, K. J. (1986). The geometry of fractals sets. Cambridge: Cambridge University Press.Google Scholar
  19. Gilboa, I. (1987). Expected utility theory with purely subjective non-additive probabilities. Journal of Mathematical Economics, 18, 141–153.CrossRefGoogle Scholar
  20. Mayag, B., Gabrish, M., & Labreuche, C. (2011). A represenation of preferences by the Choquet integral with respect to a 2-additive capacity, theory and decision (p. 71). Berlin: Springer.Google Scholar
  21. Miranda, E., Zaffalon, M., & de Cooman, G. (2012). Conglomerable natural extensions. International Journal of Approximate Reasoning, 53(8), 1200–1227.CrossRefGoogle Scholar
  22. Regazzini, E. (1985). Finitely additive conditional probabilities. Rendiconti del Seminario Matematico e Fisico di Milano, 55, 69–89.CrossRefGoogle Scholar
  23. Regazzini, E. De. (1987). Finetti’s coherence and statistical inference. The Annals of Statistics, 15(2), 845–864.CrossRefGoogle Scholar
  24. Rogers, C. A. (1970). Hausdorff measures. Cambridge: Cambridge University Press.Google Scholar
  25. Savage, L. J. (1954). The foundations of statistics. Cambridge: Cambridge University Press.Google Scholar
  26. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.CrossRefGoogle Scholar
  27. Seidenfeld, T., Schervish, M. J., & Kadane, J. K. (1995). A Representation of Partially Ordered Preferences. The Annals of Statistics, 23(6), 2168–2217.CrossRefGoogle Scholar
  28. Seidenfeld, T., Schervish, M. J., & Kadane, J. K. (1998). Non-conglomerability for finite-valued, finitely additive probability. The Indian Journal of Statistics, Special issue on Bayesian Analysis, 60(Series A), 476–491.Google Scholar
  29. Walley, P. (1981). Coherent lower (and upper) probabilities. Statistics Research Report, University of Warwick.Google Scholar
  30. Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Engineering and GeologyUniversity G. D’AnnunzioChieti-PescaraItaly

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