Open Access
2013 Empirical Bayes scaling of Gaussian priors in the white noise model
B. T. Szabó, A. W. van der Vaart, J. H. van Zanten
Electron. J. Statist. 7: 991-1018 (2013). DOI: 10.1214/13-EJS798


The performance of nonparametric estimators is heavily dependent on a bandwidth parameter. In nonparametric Bayesian methods this parameter can be specified as a hyperparameter of the nonparametric prior. The value of this hyperparameter may be made dependent on the data. The empirical Bayes method is to set its value by maximizing the marginal likelihood of the data in the Bayesian framework. In this paper we analyze a particular version of this method, common in practice, that the hyperparameter scales the prior variance. We characterize the behavior of the random hyperparameter, and show that a nonparametric Bayes method using it gives optimal recovery over a scale of regularity classes. This scale is limited, however, by the regularity of the unscaled prior. While a prior can be scaled up to make it appropriate for arbitrarily rough truths, scaling cannot increase the nominal smoothness by much. Surprisingy the standard empirical Bayes method is even more limited in this respect than an oracle, deterministic scaling method. The same can be said for the hierarchical Bayes method.


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B. T. Szabó. A. W. van der Vaart. J. H. van Zanten. "Empirical Bayes scaling of Gaussian priors in the white noise model." Electron. J. Statist. 7 991 - 1018, 2013.


Published: 2013
First available in Project Euclid: 15 April 2013

zbMATH: 1336.62039
MathSciNet: MR3044507
Digital Object Identifier: 10.1214/13-EJS798

Primary: 62G05 , 62G15
Secondary: 62G20

Keywords: Adaptation , bandwidth , Gaussian white noise , hyper-rectangle , normal means model , rate of contraction

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society

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