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On adaptive posterior concentration rates

Hoffmann, Marc; Rousseau, Judith; Schmidt-Hieber, Johannes (2015), On adaptive posterior concentration rates, Annals of Statistics, 43, 5, p. 2259-2295. 10.1214/15-AOS1341

Type
Article accepté pour publication ou publié
External document link
https://arxiv.org/abs/1305.5270v3
Date
2015
Journal name
Annals of Statistics
Volume
43
Number
5
Publisher
IMS
Pages
2259-2295
Publication identifier
10.1214/15-AOS1341
Metadata
Show full item record
Author(s)
Hoffmann, Marc

Rousseau, Judith

Schmidt-Hieber, Johannes
Abstract (EN)
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 L2 or L∞ 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.
Subjects / Keywords
Bayesian nonparametrics; minimax adaptive estimation; posterior concentration rates; sup-norm; rates of convergence

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