Titlebook: Bayes Theory; J. A. Hartigan Book 1983 Springer-Verlag New York Inc. 1983 Bayessches Verfahren.Finite.Logic.boundary element method.condit

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Conditional Probability,Kolmogorov’s exquisite formalization of conditional probability in the unitary case (1933) does not readily generalize to non-unitary probabilities. Stone and Dawid (1972) show one type of difficulty with their marginalization paradoxes for improper priors.

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Making Probabilities,The essence of Bayes theory is giving probability values to bets. Methods of generating such probabilities are what separate the various theories.

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Exponential Families,Let . be a probability on ., and choose a unitary probability . on . to minimize the information .(log(.)) subject to ....., . 1,..., .. The optimal probability . has density

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Many Normal Means,Given X, suppose ., and the .. are independent. The straight estimate .. of .. is least squares, maximum likelihood, of minimum variance among unbiased estimators, posterior means with respect to the Jeffreys density (the ..,..., .. are uniform) but for all these virtues inadmissible with loss function .for . > 2, Stein (1956).

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Springer Series in Statisticshttp://image.papertrans.cn/b/image/181817.jpg

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Ana Ferreira,Lynne Coventry,Gabriele LenziniFor example, if we assume that a penny has probability 1/2 of coming down heads on each toss, and that the different tosses are independent, it follows that the limiting proportion of heads will be 1/2 almost surely.