On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures

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.

Published in at http://dx.doi.org/10.1214/14-AOS1246 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• Rates of contraction of posterior distributions based on Gaussian process priors
• Reproducing kernel Hilbert spaces of Gaussian priors
• Frequentist coverage of adaptive nonparametric Bayesian credible sets
• Confidence bands in density estimation
• Multiscale inference about a density
• Uniform limit theorems for wavelet density estimators
• Quantile pyramids for Bayesian nonparametrics
• The Bernstein--von Mises theorem for the proportional hazard model

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