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Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited

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  • Matthew O. Jackson
  • Ehud Kalai
  • Rann Smorodinsky

Abstract

A probability distribution governing the evolution of a stochastic process has infinitely many Bayesian representations of the form mu = integral operator [subscript theta] mu[subscript theta] delta lambda (theta). Among these, a natural representation is one whose components (mu[subscript theta]'s) are 'learnable' (one can approximate mu[subscript theta] by conditioning mu on observation of the process) and 'sufficient for prediction' (mu[subscript theta]'s predictions are not aided by conditioning on observation of the process). The authors show the existence and uniqueness of such a representation under a suitable asymptotic mixing condition on the process. This representation can be obtained by conditioning on the tail-field of the process, and any learnable representation that is sufficient for prediction is asymptotically like the tail-field representation. This result is related to the celebrated de Finetti theorem, but with exchangeability weakened to an asymptotic mixing condition, and with his conclusion of a decomposition into i.i.d. component distributions weakened to components that are learnable and sufficient for prediction.

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Bibliographic Info

Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 67 (1999)
Issue (Month): 4 (July)
Pages: 875-894

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Handle: RePEc:ecm:emetrp:v:67:y:1999:i:4:p:875-894

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References

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  1. Aumann, Robert J. & Heifetz, Aviad, 2001. "Incomplete Information," Working Papers 1124, California Institute of Technology, Division of the Humanities and Social Sciences.
  2. Kalai, Ehud & Lehrer, Ehud, 1994. "Weak and strong merging of opinions," Journal of Mathematical Economics, Elsevier, vol. 23(1), pages 73-86, January.
  3. Kalai, Ehud & Lehrer, Ehud, 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Econometric Society, vol. 61(5), pages 1019-45, September.
  4. Kandori, Michihiro & Mailath, George J & Rob, Rafael, 1993. "Learning, Mutation, and Long Run Equilibria in Games," Econometrica, Econometric Society, vol. 61(1), pages 29-56, January.
  5. Jackson, Matthew O. & Kalai, Ehud, 1997. "Social Learning in Recurring Games," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 102-134, October.
  6. Dov Samet, 1996. "Common Priors and Markov Chains," Game Theory and Information 9610008, EconWPA.
  7. Rothschild, Michael, 1974. "A two-armed bandit theory of market pricing," Journal of Economic Theory, Elsevier, vol. 9(2), pages 185-202, October.
  8. Lehrer, Ehud & Smorodinsky, Rann, 1997. "Repeated Large Games with Incomplete Information," Games and Economic Behavior, Elsevier, vol. 18(1), pages 116-134, January.
  9. Dov Samet, 1996. "Looking Backwards, Looking Inwards: Priors and Introspection," Game Theory and Information 9610007, EconWPA.
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Blog mentions

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  1. Bayesian consistency
    by Eran in The Leisure of the Theory Class on 2014-08-09 16:07:10
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Cited by:
  1. Beker, Pablo & Subir Chattopadhyay, 2009. "Consumption Dynamics in General Equilibrium : A Characterisation when Markets are Incomplete," The Warwick Economics Research Paper Series (TWERPS) 921, University of Warwick, Department of Economics.
  2. Mario Gilli, 2002. "Rational Learning in Imperfect Monitoring Games," Working Papers 46, University of Milano-Bicocca, Department of Economics, revised Mar 2002.
  3. Al-Najjar, Nabil I. & Sandroni, Alvaro & Smorodinsky, Rann & Weinstein, Jonathan, 2010. "Testing theories with learnable and predictive representations," Journal of Economic Theory, Elsevier, vol. 145(6), pages 2203-2217, November.
  4. Peter Hammond & Yeneng Sun, 2001. "Monte Carlo Simulation of Macroeconomic Risk with a Continuum of Agents: The Symmetric Case," Working Papers 01015, Stanford University, Department of Economics.
  5. Dean Foster & Rakesh Vohra, 2011. "Calibration: Respice, Adspice, Prospice," Discussion Papers 1537, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  6. John H. Nachbar, 2003. "Beliefs in Repeated Games," ISER Discussion Paper 0597, Institute of Social and Economic Research, Osaka University.
  7. Hansen, Lars Peter & Sargent, Thomas J. & Turmuhambetova, Gauhar & Williams, Noah, 2006. "Robust control and model misspecification," Journal of Economic Theory, Elsevier, vol. 128(1), pages 45-90, May.
  8. Al-Najjar, Nabil & Sandroni, Alvaro, 2013. "A difficulty in the testing of strategic experts," Mathematical Social Sciences, Elsevier, vol. 65(1), pages 5-9.
  9. Turdaliev, Nurlan, 2002. "Calibration and Bayesian learning," Games and Economic Behavior, Elsevier, vol. 41(1), pages 103-119, October.
  10. Daron Acemoglu & Victor Chernozhukov & Muhamet Yildiz, 2006. "Learning and Disagreement in an Uncertain World," NBER Working Papers 12648, National Bureau of Economic Research, Inc.

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