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A representation of partially ordered preferences. (English) Zbl 0871.62008
Summary: This essay considers decision-theoretic foundations for robust Bayesian statistics. We modify the approaches of Ramsey, de Finetti, Savage and Anscombe and Aumann in giving axioms for a theory of robust preferences. We establish that preferences which satisfy axioms for robust preferences can be represented by a set of expected utilities. In the presence of two axioms relating to state-independent utility, robust preferences are represented by a set of probability/utility pairs, where the utilities are almost state-independent (in a sense which we make precise). Our goal is to focus on preference alone and to extract whatever probability and/or utility information is contained in the preference relation when that is merely a partial order. This is in contrast with the usual approach to Bayesian robustness that begins with a class of “priors” or “likelihoods”, and a single loss function, in order to derive preferences from these probability/utility assumptions.

62A01 Foundations and philosophical topics in statistics
62C05 General considerations in statistical decision theory
62C10 Bayesian problems; characterization of Bayes procedures
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[1] ANSCOMBE, F. J. and AUMANN, R. J. 1963. A definition of subjective probability. Ann. Math. Statist. 34 199 205. Z. · Zbl 0114.07204
[2] AUMANN, R. J. 1962. Utility theory without the completeness axiom. Econometrica 30 445 462. Z. · Zbl 0121.15202
[3] AUMANN, R. J. 1964. Utility theory without the completeness axiom: a correction. Econometrica 32 210 212. Z. JSTOR: · Zbl 0133.13004
[4] BERGER, J. 1985. Statistical Decision Theory and Bayesian Analy sis, 2nd ed. Springer, New York.Z. · Zbl 0572.62008
[5] DE FINETTI, B. 1937. La prevision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincare 7 1 68. Ź. · Zbl 0017.07602
[6] DEGROOT, M. 1974. Reaching a consensus. J. Amer. Statist. Assoc. 69 118 121. Z. · Zbl 0282.92011
[7] ELLSBERG, D. 1961. Risk, ambiguity, and the Savage axioms. Quart. J. Econom. 75 643 669. Z. · Zbl 1280.91045
[8] FISHBURN, P. C. 1979. Utility Theory for Decision Making. Krieger, New York. Z. · Zbl 0213.46202
[9] FISHBURN, P. C. 1982. The Foundations of Expected Utility. Reidel, Dordrecht. Z. · Zbl 0497.90001
[10] GIRON, F. J. and RIOS, S. 1980. Quasi-Bayesian behaviour: A more realistic approach to Z decision making? In Bayesian Statistics J. M. Bernardo, M. H. DeGroot, D. V. Lindley. and A. F. M. Smith, eds. 17 38. Univ. Valencia Press. Z. · Zbl 0459.62006
[11] HARTIGAN, J. A. 1983. Bay es Theory. Springer, New York. Z. Z · Zbl 0537.62007
[12] HAUSNER, M. 1954. Multidimensional utilities. In Decision Processes R. M. Thrall, C. H.. Coombs and R. L. Davis, eds. 167 180. Wiley, New York. · Zbl 0058.13804
[13] KADANE, J. B., Ed. 1984. Robustness of Bayesian Analy sis. North-Holland, Amsterdam. Z. · Zbl 0537.62031
[14] KADANE, J. B. 1986. Progress toward a more ethical method for clinical trials. Journal of Medicine and Philosophy 11 385 404. Z.
[15] KADANE, J. B. and SEDRANSK, N. 1980. Toward a more ethical clinical trial. In Bayesian Z. Statistics J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds. 329 338. Univ. Valencia Press. Z.
[16] KANNAI, Y. 1963. Existence of a utility in infinite dimensional partially ordered spaces. Israel J. Math. 1 229 234. Z. · Zbl 0152.38404
[17] LEVI, I. 1974. On indeterminate probabilities. J. Philos. 71 391 418. Z.
[18] LEVI, I. 1980. The Enterprise of Knowledge. MIT Press. Z.
[19] LEVI, I. 1990. Pareto unanimity and consensus. J. Philos. 87 481 492. Z. Z.
[20] MOSKOWITZ, H., WONG, R. T. and CHU, P.-Y. 1988. Robust interactive decision-analysis RID : Concepts, methodology, and sy stem principles. Paper 948, Krannert Graduate School of Management, Purdue Univ. Z.
[21] NAU, R. F. 1992. Indeterminate probabilities on finite sets. Ann. Statist. 20 1737 1767. Z. · Zbl 0782.62006
[22] NAU, R. F. 1993. The shape of incomplete preferences. Paper 9301, The Fuqua School of Business, Duke Univ. Z.
[23] RAMSEY, F. P. 1931. Truth and probability. In The Foundations of Mathematics and Other Z. Logical Essay s R. B. Braithwaite, ed. 156 198. Kegan, Paul, Trench, Trubner and Co. Ltd., London. Z.
[24] RIOS INSUA, D. 1990. Sensitivity Analy sis in Multi-Objective Decision Making. Springer, New York.Z.
[25] RIOS INSUA, D. 1992. On the foundations of decision making under partial information. Theory and Decision 33 83 100. Z. · Zbl 0756.90004
[26] SAVAGE, L. J. 1954. The Foundations of Statistics. Wiley, New York. Z. · Zbl 0055.12604
[27] SCHERVISH, M. J., SEIDENFELD, T. and KADANE, J. B. 1990. State-dependent utilities. J. Amer. Statist. Assoc. 85 840 847. Z. JSTOR: · Zbl 0726.90011
[28] SCHERVISH, M. J., SEIDENFELD, T. and KADANE, J. B. 1991. Shared preferences and statedependent utilities. Management Sci. 37 1575 1589. Z. · Zbl 0747.90029
[29] SEIDENFELD, T., KADANE, J. B. and SCHERVISH, M. J. 1989. On the shared preferences of two Bayesian decision makers. J. Philos. 86 225 244. Z. JSTOR:
[30] SEIDENFELD, T. and SCHERVISH, M. J. 1983. A conflict between finite additivity and avoiding Dutch Book. Philos. Sci. 50 398 412. Z. JSTOR:
[31] SEIDENFELD, T., SCHERVISH, M. J. and KADANE, J. B. 1990. Decisions without ordering. In Z. Acting and Reflecting W. Sieg, ed. 143 170. Kluwer, Dordrecht. Z.
[32] SMITH, C. A. B. 1961. Consistency in statistical inference and decision. J. Roy. Statist. Soc. Ser. B 23 1 25. Z. JSTOR: · Zbl 0124.09603
[33] SZPILRAJN, E. 1930. Sur l’extension de l’ordre partiel. Fund. Math. 16 386 389. Z. · JFM 56.0843.02
[34] VON NEUMANN, J. and MORGENSTERN, O. 1947. Theory of Games and Economic Behavior, 2nd ed. Princeton Univ. Press. Z. · Zbl 1241.91002
[35] WALLEY, P. 1991. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London. Z. · Zbl 0732.62004
[36] WHITE, C. C. 1986. A posteriori representations based on linear inequality descriptions of a priori and conditional probabilities. IEEE Trans. Sy stems Man Cy bernet. 16 570 573. Z.
[37] WILLIAMS, P. 1976. Indeterminate probabilities. In Formal Methods in the Methodology of Z. Empirical Sciences M. Przelecki, K. Szaniawski and R. Wojcicki, eds. 229 246. Reidel, Dordrecht. · Zbl 0395.60003
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