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The Generalization of de Finetti's Representation Theorem to Stationary Probabilities

Jan von Plato
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association
Vol. 1982, Volume One: Contributed Papers (1982), pp. 137-144
Stable URL: http://www.jstor.org/stable/192662
Page Count: 8
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The Generalization of de Finetti's Representation Theorem to Stationary Probabilities
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Abstract

de Finetti's representation theorem of exchangeable probabilities as unique mixtures of Bernoullian probabilities is a special case of a result known as the ergodic decomposition theorem. It says that stationary probability measures are unique mixtures of ergodic measures. Stationarity implies convergence of relative frequencies, and ergodicity the uniqueness of limits. Ergodicity therefore captures exactly the idea of objective probability as a limit of relative frequency (up to a set of measure zero), without the unnecessary restriction to probabilistically independent events as in de Finetti's theorem. The ergodic decomposition has in some applications to dynamical systems a physical content, and de Finetti's reductionist interpretation of his result is not adequate in these cases.

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