Abstract
When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.
Similar content being viewed by others
REFERENCES
Berger, J. (1990), Robust Bayesian analysis: sensitivity to the prior, Journal of Statistical Planning and Inference, 25, 303–328.
Brown, P.M., (1976) Conditionalization and expected utility, Philosophy of Science 43, 415–419.
Camerer, C.F. and Weber M. (1941), Recent developments in modelling preferences: uncertainty and ambiguity, Journal of Risk and Uncertainty, 5, 325–370.
Choquet, G. (1953–1954), Theory of capacities, Annals lÍnstitut Fourier, 5, 131–295.
Cohen, M. and J.Y. Jaffray, (1985), Decision making in a case of mixed uncertainty: a normative model, Journal of Mathematical Psychology 29, 428–442.
Dempster, A.P. (1967), Upper and lower probabilities induced by a multivalued map, Annals of Mathematical Statistics, 38, 325–339.
Dempster, A.P. (1968), A generalization of Bayesian inference, Journal of Royal Statistics Society, Ser. B, 30, 205–247.
Dunford, N. and J.T. Schwartz, Linear Operators, Part 1, Interscience, New York, 1957.
Ellsberg, D. (1961), Risk, ambiguity and the Savage axioms, Quarterly Journal of Economics, 75, 643–669.
Fagin, R. and Halpern, J. (1989), A new approach to updating belief, in Proc. 6th Conference Uncertainty and AI, 1990. I.B.M. Res. Rep. RJ 7222.
Gilboa, I. (1987), Expected utility theory without purely subjective non-additive probabilities, Journal of Mathematical Economics 16, 65–88.
Gilboa, I. (1989), Additivizations of non-additive measures, Mathematical Operations Research, 14, 1–17.
Gilboa, I. and Schmeidler D. (1989), Maxmin expected utility with non-unique prior, Journal of Mathematical Economics, 18, 141–153.
Gilboa, I. and Schmeidler D. (1993), Updating ambiguous beliefs, Journal of Economic Theory, 59, 33–49.
Jaffray, J.Y. (1992), Bayesian updating and belief Functions, IEEE Transactions on Systems, Man and Cybernetics, 22, 1144–1152.
Jaffray, J.Y. (1994), Dynamic decision making with belief functions, in Yager et al. (ed.), Advances in the Dempster-Shafer Theory of Evidence, Wiley.
Kreps, D. (1998), Notes on the Theory of Choice. Westview, Boulder.
Savage, L.J. (1954), The Foundations of Statistics. Wiley, New York.
Schmeidler, D. (1982), Subjective probability without additivity, Working Paper, Foerder Institute for Economic Research, Tel Aviv University.
Schmeidler, D. (1989), Subjective probability and expected utility without additivity, Econometrica, 57, 571–587.
Shafer, G. (1976), A Mathematical Theory of Evidence, Princeton, NJ: Princeton University Press.
Wakker, P. (1989), Continuous subjective expected utility with non-additive probabilities, Journal of Mathematical Economics, 18, 1–17.
Wald, A. (1950), Statistical Decision Functions. Wiley, New York.
Walley, P. (1981), Coherent lower (and upper) probabilities, Statistics research report, University of Warwick, Coventry.
Walley, P. (1991), Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London.
Wasserman, L.A. and Kadane J. (1990), Bayes' theorem for Choquet capacities, The Annals of Statistics, 18, 1328–1339.
Wasserman, L., Lavine M., and Wolpert R. (1993), Linearization of Bayesian robustness problems, J. Statistical Planning and Inference, 37, 307–316.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pacheco Pires, C. A Rule For Updating Ambiguous Beliefs. Theory and Decision 53, 137–152 (2002). https://doi.org/10.1023/A:1021255808323
Issue Date:
DOI: https://doi.org/10.1023/A:1021255808323