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On consistency and null sets in Bayes estimation Delbrouck, Lucien Elie Nicolas

Abstract

A basic result of Doob states that, under very weak measurability assumptions, Bayes’ estimators are consistent for almost all parameter points. First it is shown that even when this exceptional set is finite, the effect of putting positive prior mass on each point of the set may result in creating a new exceptional set, larger than the original one, rather than in eliminating the lack of consistency. The .posterior densities are then studied and it is shown that under fairly strong regularity conditions the corresponding posterior distributions tend, in the limit, to concentrate their mass on a particular point in the parameter set. If in addition, distinct parameter points correspond to distinct probability measures, then it is shown that both the maximum likelihood and the Bayes' estimators are consistent for all parameter values.

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