Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Bayesian modelization of sparse sequences and maxisets for Bayes rules


Code(s) de Classification MSC:

Résumé: In this paper, our aim is to estimate sparse sequences in the framework of the heteroscedastic white noise model. To modelize sparsity, we consider a Bayesian model composed of a mixture of a heavy-tailed density and a point mass at zero. To evaluate the performance of the Bayes rules (the median or the mean of the posterior distribution), we exploit an alternative to the minimax setting developed in particular by Kerkyacharian and Picard: we evaluate the maxisets for each of these estimators. Using this approach, we compare the performance of Bayesian procedures with thresholding ones. Furthermore, the maxisets obtained can be viewed as weighted versions of weak $l_q$ spaces that naturally modelize sparsity. This remark leads us to investigate the following problem: how can we choose the prior parameters to build typical realizations of weak $l_q$ spaces ?

Mots Clés: Bayes rules ; Bayesian model ; heteroscedastic white noise model ; maxisets ; rate of convergence ; sparsity ; thresholding rules ; weak $l_q$ spaces

Date: 2002-06-26

Prépublication numéro: PMA-741

Pdf file : PMA-741.pdf