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Extending Doob’s consistency theorem to nonparametric densities. (English) Zbl 1055.62053
From the paper: J. L. Doob [Colloques Intnat. Centre Nat. Rech. Sci. 13, Calcul des Probabilités et ses applications, 23–27 (1999; Zbl 0041.45101)] gave an application of the martingale convergence theorem to the study of consistency of Bayesian procedures. In particular, it is proved that if there exists a consistent estimator of a parameter \(\widetilde\theta\), then the posterior distribution of \(\widetilde\theta\) accumulates in neighbourhoods of \(\widetilde\theta\) almost surely.
We extend Doob s well known result on Bayesian consistency. The extension covers the case where the nonparametric prior is fully supported by densities. However, our use of martingales differs from that of Doob. We also consider rates.

MSC:
62G20 Asymptotic properties of nonparametric inference
60G42 Martingales with discrete parameter
60F15 Strong limit theorems
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