Open Access
October 2015 On adaptive posterior concentration rates
Marc Hoffmann, Judith Rousseau, Johannes Schmidt-Hieber
Ann. Statist. 43(5): 2259-2295 (October 2015). DOI: 10.1214/15-AOS1341


We investigate the problem of deriving posterior concentration rates under different loss functions in nonparametric Bayes. We first provide a lower bound on posterior coverages of shrinking neighbourhoods that relates the metric or loss under which the shrinking neighbourhood is considered, and an intrinsic pre-metric linked to frequentist separation rates. In the Gaussian white noise model, we construct feasible priors based on a spike and slab procedure reminiscent of wavelet thresholding that achieve adaptive rates of contraction under $L^{2}$ or $L^{\infty}$ metrics when the underlying parameter belongs to a collection of Hölder balls and that moreover achieve our lower bound. We analyse the consequences in terms of asymptotic behaviour of posterior credible balls as well as frequentist minimax adaptive estimation. Our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment. The upper bound is attained for certain sieve priors and enables to extend our results to density estimation.


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Marc Hoffmann. Judith Rousseau. Johannes Schmidt-Hieber. "On adaptive posterior concentration rates." Ann. Statist. 43 (5) 2259 - 2295, October 2015.


Received: 1 May 2013; Revised: 1 April 2015; Published: October 2015
First available in Project Euclid: 16 September 2015

zbMATH: 1327.62306
MathSciNet: MR3396985
Digital Object Identifier: 10.1214/15-AOS1341

Primary: 62G08 , 62G20
Secondary: 62G07

Keywords: Bayesian nonparametrics , minimax adaptive estimation , posterior concentration rates , rates of convergence , sup-norm

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 5 • October 2015
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