The Annals of Statistics

On adaptive posterior concentration rates

Marc Hoffmann, Judith Rousseau, and Johannes Schmidt-Hieber

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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.

Article information

Ann. Statist., Volume 43, Number 5 (2015), 2259-2295.

Received: May 2013
Revised: April 2015
First available in Project Euclid: 16 September 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties 62G08: Nonparametric regression
Secondary: 62G07: Density estimation

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


Hoffmann, Marc; Rousseau, Judith; Schmidt-Hieber, Johannes. On adaptive posterior concentration rates. Ann. Statist. 43 (2015), no. 5, 2259--2295. doi:10.1214/15-AOS1341.

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