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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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  • Matthew O. Jackson & Ehud Kalai & Rann Smorodinsky, 1998. "Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited," Discussion Papers 1228, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1228
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    1. Bayesian consistency
      by Eran in The Leisure of the Theory Class on 2014-08-09 21:07:10

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    Cited by:

    1. Peter J. Hammond & Yeneng Sun, 2003. "Monte Carlo simulation of macroeconomic risk with a continuum of agents: the symmetric case," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(2), pages 743-766, March.
    2. Levy, Yehuda John, 2015. "Limits to rational learning," Journal of Economic Theory, Elsevier, vol. 160(C), pages 1-23.
    3. Dean Foster & Rakesh Vohra, 2011. "Calibration: Respice, Adspice, Prospice," Discussion Papers 1537, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    4. Daron Acemoglu & Victor Chernozhukov & Muhamet Yildiz, 2006. "Learning and Disagreement in an Uncertain World," NBER Working Papers 12648, National Bureau of Economic Research, Inc.
    5. Yildiz, Muhamet & Acemoglu, Daron & Chernozhukov, Victor, 2016. "Fragility of asymptotic agreement under Bayesian learning," Theoretical Economics, Econometric Society, vol. 11(1), January.
    6. 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.
    7. Mario Gilli, 2002. "Rational Learning in Imperfect Monitoring Games," Working Papers 46, University of Milano-Bicocca, Department of Economics, revised Mar 2002.
    8. Pe[combining cedilla]ski, Marcin, 2011. "Prior symmetry, similarity-based reasoning, and endogenous categorization," Journal of Economic Theory, Elsevier, vol. 146(1), pages 111-140, January.
    9. John H. Nachbar, 2005. "Beliefs in Repeated Games," Econometrica, Econometric Society, vol. 73(2), pages 459-480, March.
    10. Beker, Pablo & Chattopadhyay, Subir, 2010. "Consumption dynamics in general equilibrium: A characterisation when markets are incomplete," Journal of Economic Theory, Elsevier, vol. 145(6), pages 2133-2185, November.
    11. Al-Najjar, Nabil & Sandroni, Alvaro, 2013. "A difficulty in the testing of strategic experts," Mathematical Social Sciences, Elsevier, vol. 65(1), pages 5-9.
    12. Turdaliev, Nurlan, 2002. "Calibration and Bayesian learning," Games and Economic Behavior, Elsevier, vol. 41(1), pages 103-119, October.
    13. Nabil I. Al-Najjar & Eran Shmaya, 2015. "Uncertainty and Disagreement in Equilibrium Models," Journal of Political Economy, University of Chicago Press, vol. 123(4), pages 778-808.
    14. 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.

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