Electronic Journal of Statistics

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

Haralambie Leahu

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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

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

First available in Project Euclid: 10 May 2011

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Zentralblatt MATH identifier

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]

Nonparametric Bernstein-von Mises Theorem


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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