Open Access
2011 On the Bernstein-von Mises phenomenon in the Gaussian white noise model
Haralambie Leahu
Electron. J. Statist. 5: 373-404 (2011). DOI: 10.1214/11-EJS611


We study the Bernstein-von Mises (BvM) phenomenon, i.e., Bayesian credible sets and frequentist confidence regions for the estimation error coincide asymptotically, for the infinite-dimensional Gaussian white noise model governed by Gaussian prior with diagonal-covariance structure. While in parametric statistics this fact is a consequence of (a particular form of) the BvM Theorem, in the nonparametric setup, however, the BvM Theorem is known to fail even in some, apparently, elementary cases. In the present paper we show that BvM-like statements hold for this model, provided that the parameter space is suitably embedded into the support of the prior. The overall conclusion is that, unlike in the parametric setup, positive results regarding frequentist probability coverage of credible sets can only be obtained if the prior assigns null mass to the parameter space.


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Haralambie Leahu. "On the Bernstein-von Mises phenomenon in the Gaussian white noise model." Electron. J. Statist. 5 373 - 404, 2011.


Published: 2011
First available in Project Euclid: 10 May 2011

zbMATH: 1274.62290
MathSciNet: MR2802048
Digital Object Identifier: 10.1214/11-EJS611

Primary: 62G08 , 62G20
Secondary: 28C20 , 60B12 , 60F05 , 62J05

Keywords: Nonparametric Bernstein-von Mises Theorem

Rights: Copyright © 2011 The Institute of Mathematical Statistics and the Bernoulli Society

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