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On adaptive posterior concentration rates

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Date
2015
Link to item file
https://arxiv.org/abs/1305.5270v3
Dewey
Probabilités et mathématiques appliquées
Sujet
Bayesian nonparametrics; minimax adaptive estimation; posterior concentration rates; sup-norm; rates of convergence
Journal issue
Annals of Statistics
Volume
43
Number
5
Publication date
2015
Article pages
2259-2295
Publisher
IMS
DOI
http://dx.doi.org/10.1214/15-AOS1341
URI
https://basepub.dauphine.fr/handle/123456789/16328
Collections
  • CEREMADE : Publications
Metadata
Show full item record
Author
Hoffmann, Marc
Rousseau, Judith
Schmidt-Hieber, Johannes
Type
Article accepté pour publication ou publié
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.

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