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Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited
Author(s) -
Jackson Matthew O.,
Kalai Ehud,
Smorodinsky Rann
Publication year - 1999
Publication title -
econometrica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 16.7
H-Index - 199
eISSN - 1468-0262
pISSN - 0012-9682
DOI - 10.1111/1468-0262.00055
Subject(s) - bayesian probability , representation (politics) , econometrics , mathematical economics , economics , computer science , mathematics , artificial intelligence , political science , politics , law
A probability distribution governing the evolution of a stochastic process has infinitely many Bayesian representations of the form μ =∫ μ d λ ( θ ). Among these, a natural representation is one whose components ( μ 's) are ‘learnable’ (one can approximateμ by conditioning μ on observation of the process) and ‘sufficient for prediction’ ( μ '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.