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
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1305.5270v3
Journal nameAnnals of Statistics
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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 / KeywordsBayesian nonparametrics; minimax adaptive estimation; posterior concentration rates; sup-norm; rates of convergence
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