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arXiv · 1310.2484

On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures

Ismaël Castillo·Richard Nickl
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Abstract

We continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999-2028]. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein-von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides with efficient frequentist procedures, including Donsker- and Kolmogorov-Smirnov theorems for the random posterior cumulative distribution functions. We also show that multiscale posterior credible bands for the regression or density function are optimal frequentist confidence bands.

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Ismaël Castillo, Richard Nickl. 2014-10-02. On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures. https://doi.org/10.1214/14-aos1246

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