## On the Bernstein-von Mises phenomenon in the Gaussian white noise model.(English)Zbl 1274.62290

Summary: 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.

### MSC:

 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F05 Central limit and other weak theorems 62J05 Linear regression; mixed models 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

### Keywords:

nonparametric Bernstein-von Mises theorem
Full Text:

### References:

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