Open Access
December 2017 Adaptive Bernstein–von Mises theorems in Gaussian white noise
Kolyan Ray
Ann. Statist. 45(6): 2511-2536 (December 2017). DOI: 10.1214/16-AOS1533


We investigate Bernstein–von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in $L^{2}$ and $L^{\infty}$, respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets based on the posterior distribution. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved.


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Kolyan Ray. "Adaptive Bernstein–von Mises theorems in Gaussian white noise." Ann. Statist. 45 (6) 2511 - 2536, December 2017.


Received: 1 July 2015; Revised: 1 December 2016; Published: December 2017
First available in Project Euclid: 15 December 2017

zbMATH: 1384.62158
MathSciNet: MR3737900
Digital Object Identifier: 10.1214/16-AOS1533

Primary: 62G20
Secondary: 62G08 , 62G15

Keywords: Adaptation , Bayesian inference , confidence set , Credible set , posterior asymptotics

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 6 • December 2017
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