2015/10/13 by Stéphanie van der Pas, S.L. van der Pas, Jean-Bernard Salomond +3
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Bayesian inference #Bayesian probability #Contraction (grammar) #Markov Chains and Monte Carlo Methods #Minimax #Posterior probability #Prior probability #Scale (ratio) #Shrinkage #Shrinkage estimator #Statistical Methods and Inference #math.ST #stat.TH
paper · pdf · doi:10.1214/16-ejs1130
published as Electron. J. Statist. 10 (2016), no. 1, 976--1000. http://projecteuclid.org/euclid.ejs/1460463652
arxiv created 2015/10/13 · openalex publication_date 2016/01/01 · openalex created_date 2016/06/24 · arxiv updated 2016/08/16 · openalex updated_date 2026/08/05
The first Bayesian results for the sparse normal means problem were proven for spike-and-slab priors. However, these priors are less convenient from a computational point of view. In the meanwhile, a large number of continuous shrinkage priors has been proposed. Many of these shrinkage priors can be written as a scale mixture of normals, which makes them particularly easy to implement. We propose general conditions on the prior on the local variance in scale mixtures of normals, such that posterior contraction at the minimax rate is assured. The conditions require tails at least as heavy as Laplace, but not too heavy, and a large amount of mass around zero relative to the tails, more so as the sparsity increases. These conditions give some general guidelines for choosing a shrinkage prior for estimation under a nearly black sparsity assumption. We verify these conditions for the class of priors considered in [12], which includes the horseshoe and the normal-exponential gamma priors, and for the horseshoe+, the inverse-Gaussian prior, the normal-gamma prior, and the spike-and-slab Lasso, and thus extend the number of shrinkage priors which are known to lead to posterior contraction at the minimax estimation rate.