On the generic nonconvergence of Bayesian actions and beliefs
 M. Feldman
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SupposeY _{n} is a sequence of i.i.d. random variables taking values in Y, a complete, separable, nonfinite metric space. The probability law indexed byθεΘ, is unknown to a Bayesian statistician with priorμ, observing this process. Generalizing Freedman [8], we show that “generically” (i.e., for a residual family of (θ,μ) pairs) the posterior beliefs do not weakly converge to a pointmass at the “true”θ. Furthermore, for every open setG ⊂Θ, generically, the Bayesian will attach probability arbitrarily close to one toG infinitely often. The above result is applied to a twoarmed bandit problem with geometric discounting where armk yields an outcome in a complete, separable metric spaceY _{k}. If the infimum of the possible rewards from playing armk is less than the infimum from playing armk', then armk is (generically) chosen only finitely often. If the infimum of the rewards are equal, then both arms are played infinitely often.
 Ash, R (1972) Real analysis and probability. Academic Press, New York
 Barron A (1988) The exponential convergence of posterior probabilities with implications for Bayes estimators of density functions. Working Paper, Department of Statistics, University of Illinois at UrbanaChampaign
 Berry, DA, Fristedt, B (1985) Bandit problems: sequential allocation of experiments. Chapman & Hall, London
 Bikhchandani S, Sharma S (1990) Optimal search with learning. Working Paper, UCLA
 Billingsley, P (1968) Convergence of probability measures. Wiley, New York
 Blackwell, D (1965) Discounted dynamic programming. Ann Math Stat 36: pp. 226235
 Blume, L, Easley, D (1984) Rational expectations equilibrium: an alternative approach. J Econ Theory 34: pp. 116129
 Bray M, Kreps D (1981) Rational learning and rational expectations. Mimeo, Stanford University
 Cyert, R, DeGroot, M (1974) Rational expectations and Bayesian analysis. J Polit Econ 82: pp. 521536
 Diaconis, P, Freedman, D (1986) On the consistency of Bayes estimates. Ann Stat 14: pp. 126
 Dubins, L, Freedman, D (1964) Measurable sets of measures. Pac J Math 14: pp. 12111222
 Dudley, RM (1966) Convergence of baire measures. Stud Math 27: pp. 251268
 Dudley, RM (1989) Real analysis and probability. Wadsworth & Brooks/Cole, Belmont, CA
 Dynkin, EB, Yashkevich, AA (1979) Controlled Markov processes. Springer, Berlin Heidelberg New York
 Easley, D, Kiefer, N (1988) Controlling a stochastic process with unknown parameters. Econometria 56: pp. 10451064
 Easley, D, Kiefer, N (1989) Optimal learning with endogenous data. Int Econ Rev 30: pp. 963978
 Feldman, M (1987) Bayesian learning and convergence to rational expectations. J Math Econ 16: pp. 297313
 Feldman, M (1987) An example of convergence of rational expectations with heterogeneous beliefs. Int Econ Rev 00: pp. 000000
 Feldman, M, McLennan, A (1989) Learning in a repeated statistical decision problem with normal disturbances 000–000: pp. 000000
 Freedman, D (1963) Asymptotic behavior of Bayes estimates in the discrete case. Ann Math Stat 34: pp. 13861403
 Freedman, D (1965) On the asymptotic behavior of Bayes estimates in the discrete case II. Ann Math Stat 36: pp. 454456
 Gittens, J, Jones, D (1974) A dynamic allocation index for the sequential design of experiments. Progress in Statistics. NorthHolland, Amsterdam
 Kelley, J (1985) General topology. Springer, Berlin Heidelberg New York
 Kiefer, N (1989) Optimal collection of information by partially informed agents. Econ Rev 7: pp. 113148
 Kiefer, N, Nyarko, Y (1988) Control of a linear regression process with unknown parameters. Third International Symposium in Economic Theory and Econometrics. Cambridge University Press, Cambridge
 Kiefer, N, Nyarko, Y (1989) Optimal control of an unknown linear process with learning. Int Econ Rev 30: pp. 571588
 Maitra, A (1968) Discounted dynamic programming in compact metric spaces. Sankhya [Ser A] 30: pp. 211216
 McLennan A (1987) Incomplete learning in a repeated statistical decision problem. Working Paper, University of Minnesota
 Oxtoby, J (1980) Measure and category. Springer, Berlin Heidelberg New York
 Parthasarathy, KR (1967) Probability measures on metric spaces. Academic Press, New York
 Reider, U (1975) Bayesian dynamic programming. Adi Appl Probab 7: pp. 330348
 Ross, S (1983) Introduction to stochastic dynamic programming. Academic Press, New York
 Rothschild, M (1974) A twoarmed bandit theory of marrket pricing 9: pp. 185202
 Royden, HL (1988) Real analysis. Macmillan, New York
 Schwartz, L (1965) On Bayes' procedures 4: pp. 1026
 Strasser, H (1985) Mathematical theory of statistics. Springer, Berlin Heidelberg New York Townsend
 Townsend, R (1978) Market anticipations, rational expectations and Bayesian analysis. Int Econ Rev 19: pp. 481494
 Whittle, P (1982) Optimization over time: dynamic programming and stochastic control (I). Wiley, New York
 Title
 On the generic nonconvergence of Bayesian actions and beliefs
 Journal

Economic Theory
Volume 1, Issue 4 , pp 301321
 Cover Date
 19911201
 DOI
 10.1007/BF01229311
 Print ISSN
 09382259
 Online ISSN
 14320479
 Publisher
 SpringerVerlag
 Additional Links
 Topics
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 Authors

 M. Feldman ^{(1)}
 Author Affiliations

 1. Department of Economics, University of Illinois, 61820, UrbanaChampaign, IL, USA