The Annals of Statistics

Adaptive Bernstein–von Mises theorems in Gaussian white noise

Kolyan Ray

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Abstract

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.

Article information

Source
Ann. Statist., Volume 45, Number 6 (2017), 2511-2536.

Dates
Received: July 2015
Revised: December 2016
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1513328581

Digital Object Identifier
doi:10.1214/16-AOS1533

Mathematical Reviews number (MathSciNet)
MR3737900

Zentralblatt MATH identifier
1384.62158

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G15: Tolerance and confidence regions 62G08: Nonparametric regression

Keywords
Bayesian inference posterior asymptotics adaptation credible set confidence set

Citation

Ray, Kolyan. Adaptive Bernstein–von Mises theorems in Gaussian white noise. Ann. Statist. 45 (2017), no. 6, 2511--2536. doi:10.1214/16-AOS1533. https://projecteuclid.org/euclid.aos/1513328581


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Supplemental materials

  • Supplement to “Adaptive Bernstein–von Mises theorems in Gaussian white noise”. All proofs together with additional results are given in the Supplement [39].