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October 2014 On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures
Ismaël Castillo, Richard Nickl
Ann. Statist. 42(5): 1941-1969 (October 2014). DOI: 10.1214/14-AOS1246


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. "On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures." Ann. Statist. 42 (5) 1941 - 1969, October 2014.


Published: October 2014
First available in Project Euclid: 11 September 2014

zbMATH: 1305.62190
MathSciNet: MR3262473
Digital Object Identifier: 10.1214/14-AOS1246

Primary: 62G20
Secondary: 62G08 , 62G15

Keywords: Bayesian inference , multiscale statistics , posterior asymptotics

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 5 • October 2014
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