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Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited

  • Matthew O. Jackson
  • Ehud Kalai
  • Rann Smorodinsky

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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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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  1. Ehud Kalai & Ehud Lehrer, 1990. "Rational Learning Leads to Nash Equilibrium," Discussion Papers 895, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  2. Aumann, Robert J. & Heifetz, Aviad, 2002. "Incomplete information," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 43, pages 1665-1686 Elsevier.
  3. Dov Samet, 1996. "Common Priors and Markov Chains," Game Theory and Information 9610008, EconWPA.
  4. Kalai, Ehud & Lehrer, Ehud, 1994. "Weak and strong merging of opinions," Journal of Mathematical Economics, Elsevier, vol. 23(1), pages 73-86, January.
  5. M. Kandori & G. Mailath & R. Rob, 1999. "Learning, Mutation and Long Run Equilibria in Games," Levine's Working Paper Archive 500, David K. Levine.
  6. Rothschild, Michael, 1974. "A two-armed bandit theory of market pricing," Journal of Economic Theory, Elsevier, vol. 9(2), pages 185-202, October.
  7. Matthew Jackson & Ehud Kalai, 1995. "Social Learning in Recurring Games," Discussion Papers 1138, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  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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