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Foundations of neo-Bayesian statistics

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  • Amarante, Massimiliano

Abstract

We study an axiomatic model of preferences, which contains as special cases Subjective Expected Utility, Choquet Expected Utility, Maxmin and Maxmax Expected Utility and many other models. First, we give a complete characterization of the class of functionals representing these preferences. Then, we show that any such functional can be represented as a Choquet integral where is the canonical mapping from the space of bounded [Sigma]-measurable functions into the space of weak*-continuous affine functions on a weak*-compact, convex set of probability measures on [Sigma]. Conversely, any preference relation defined by means of such functionals satisfies the axioms of the model we study. Different properties of the capacity give rise to different models. Our result shows that the idea of Choquet integration is general enough to embrace all the models mentioned above. In doing so, it widens the range of applicability of well-known procedures in robust statistics theory such as the Neyman-Pearson lemma for capacities [P.J. Huber, V. Strassen, Minimax tests and the Neyman-Pearson lemma for capacities, Ann. Statist. 1 (1973) 251-263], Bayes' theorem for capacities [J.B. Kadane, L. Wasserman, Bayes' theorem for Choquet capacities, Ann. Statist. 18 (1990) 1328-1339] or of results like the Law of Large numbers for capacities [F. Maccheroni, M. Marinacci, A strong law of large numbers for capacities, Ann. Probab. 33 (2005) 1171-1178].

Suggested Citation

  • Amarante, Massimiliano, 2009. "Foundations of neo-Bayesian statistics," Journal of Economic Theory, Elsevier, vol. 144(5), pages 2146-2173, September.
  • Handle: RePEc:eee:jetheo:v:144:y:2009:i:5:p:2146-2173
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    References listed on IDEAS

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    1. Paolo Ghirardato & Massimo Marinacci, 2001. "Risk, Ambiguity, and the Separation of Utility and Beliefs," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 864-890, November.
    2. Ghirardato, Paolo & Maccheroni, Fabio & Marinacci, Massimo, 2004. "Differentiating ambiguity and ambiguity attitude," Journal of Economic Theory, Elsevier, vol. 118(2), pages 133-173, October.
    3. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
    4. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    5. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    6. Massimo Marinacci, 2002. "Learning from ambiguous urns," Statistical Papers, Springer, vol. 43(1), pages 143-151, January.
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    Cited by:

    1. repec:gam:jrisks:v:4:y:2016:i:1:p:8:d:66161 is not listed on IDEAS
    2. Peter Klibanoff & Massimo Marinacci & Sujoy Mukerji, 2012. "On the Smooth Ambiguity Model: A Reply," Econometrica, Econometric Society, vol. 80(3), pages 1303-1321, May.
    3. Massimiliano Amarante, 2016. "A representation of risk measures," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 39(1), pages 95-103, April.
    4. Ghossoub, Mario, 2011. "Monotone equimeasurable rearrangements with non-additive probabilities," MPRA Paper 37629, University Library of Munich, Germany, revised 23 Mar 2012.
    5. Peter Klibanoff & Sujoy Mukerji & Kyoungwon Seo, 2014. "Perceived Ambiguity and Relevant Measures," Econometrica, Econometric Society, vol. 82, pages 1945-1978, September.
    6. Massimo Marinacci, 2015. "Model Uncertainty," Journal of the European Economic Association, European Economic Association, vol. 13(6), pages 1022-1100, December.
    7. Cerreia-Vioglio, Simone & Maccheroni, Fabio & Marinacci, Massimo & Montrucchio, Luigi, 2013. "Ambiguity and robust statistics," Journal of Economic Theory, Elsevier, vol. 148(3), pages 974-1049.
      • Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2011. "Ambiguity and Robust Statistics," Working Papers 382, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    8. Massimiliano Amarante, 2017. "Conditional expected utility," Theory and Decision, Springer, vol. 83(2), pages 175-193, August.
    9. Dumav, Martin & Stinchcombe, Maxwell B., 2014. "The von Neumann/Morgenstern approach to ambiguity," Center for Mathematical Economics Working Papers 480, Center for Mathematical Economics, Bielefeld University.
    10. AMARANTE, Massimiliano, 2009. "Toward a Rational-Choice Foundation of Non-Additive Theories," Cahiers de recherche 13-2009, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    11. Giraud, Raphaël, 2014. "Second order beliefs models of choice under imprecise risk: non-additive second order beliefs vs. nonlinear second order utility," Theoretical Economics, Econometric Society, vol. 9(3), September.
    12. Karni, Edi & Maccheroni, Fabio & Marinacci, Massimo, 2015. "Ambiguity and Nonexpected Utility," Handbook of Game Theory with Economic Applications, Elsevier.
    13. AMARANTE, Massimiliano, 2014. "What is ambiguity?," Cahiers de recherche 2014-01, Universite de Montreal, Departement de sciences economiques.
    14. Gumen, Anna & Savochkin, Andrei, 2013. "Dynamically stable preferences," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1487-1508.
    15. Massimiliano Amarante, 2015. "Analogy in Decision Making," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 1027-1041, October.
    16. Massimiliano Amarante & Mario Ghossoub, 2016. "Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer," Risks, MDPI, Open Access Journal, vol. 4(1), pages 1-27, March.
    17. Johanna Etner & Meglena Jeleva & Jean-Marc Tallon, 2009. "Decision theory under uncertainty," Post-Print halshs-00429573, HAL.

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