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Bayesian Representation of Stochastic Processes Under Learning: De Finetti Revisited

Matthew O. Jackson, Ehud Kalai and Rann Smorodinsky
Econometrica
Vol. 67, No. 4 (Jul., 1999), pp. 875-893
Published by: Econometric Society
Stable URL: http://www.jstor.org/stable/2999460
Page Count: 19
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Bayesian Representation of Stochastic Processes Under Learning: De Finetti Revisited
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Abstract

A probability distribution governing the evolution of a stochastic process has infinitely many Bayesian representations of the form $\mu =\int_{\Theta}\mu _{\theta }d\lambda (\theta)$. Among these, a natural representation is one whose components $(\mu _{\theta}\text{'}{\rm s})$ are "learnable" (one can approximate μ θ by conditioning μ on observation of the process) and "sufficient for prediction" ($\mu _{\theta}\text{'}{\rm s}$ predictions are not aided by conditioning on observation of the process). We 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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