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December 2011 Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞
Evarist Giné, Richard Nickl
Ann. Statist. 39(6): 2883-2911 (December 2011). DOI: 10.1214/11-AOS924


The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking Lr-norm neighborhoods, 1 ≤ r ≤ ∞, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for 1 ≤ r ≤ 2, but deteriorate as r increases beyond 2. In the case of Gaussian nonparametric regression a Gaussian prior is devised for which the posterior contracts at the optimal rate in all Lr-norms, 1 ≤ r ≤ ∞.


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Evarist Giné. Richard Nickl. "Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞." Ann. Statist. 39 (6) 2883 - 2911, December 2011.


Published: December 2011
First available in Project Euclid: 24 January 2012

zbMATH: 1246.62095
MathSciNet: MR3012395
Digital Object Identifier: 10.1214/11-AOS924

Primary: 62G20
Secondary: 62G07 , 62G08

Keywords: nonparametric hypothesis testing , Posterior , rate of contraction

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 6 • December 2011
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