Electronic Journal of Statistics

On the Bernstein-von Mises phenomenon in the Gaussian white noise model

Haralambie Leahu

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 373-404.

Dates
First available in Project Euclid: 10 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1305034907

Digital Object Identifier
doi:10.1214/11-EJS611

Mathematical Reviews number (MathSciNet)
MR2802048

Zentralblatt MATH identifier
1274.62290

Subjects
Primary: 62G08: Nonparametric regression 62G20: Asymptotic properties
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F05: Central limit and other weak theorems 62J05: Linear regression 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
Nonparametric Bernstein-von Mises Theorem

Citation

Leahu, Haralambie. On the Bernstein-von Mises phenomenon in the Gaussian white noise model. Electron. J. Statist. 5 (2011), 373--404. doi:10.1214/11-EJS611. https://projecteuclid.org/euclid.ejs/1305034907


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