Bayesian Modelling of Sparse Sequences and Maxisets for Bayes Rules

In this paper, our aim is to estimate sparse sequences in the framework of the heteroscedastic white noise model. To model 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 determine 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 lq spaces that naturally model sparsity. This remark leads us to investigate the following problem: how can we choose the prior parameters to build typical realizations of weighted weak lq spaces?