TY - JOUR
T1 - Conditions for posterior contraction in the sparse normal means problem
AU - van der Pas, S. L.
AU - Salomond, J. B.
AU - Schmidt-Hieber, J.
N1 - Publisher Copyright: © 2016, Institute of Mathematical Statistics. All rights reserved. Copyright: Copyright 2016 Elsevier B.V., All rights reserved.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
KW - Bayesian inference
KW - Frequentist Bayes
KW - Horseshoe
KW - Horseshoe+
KW - Nearly black vectors
KW - Normal means problem
KW - Posterior contraction
KW - Shrinkage priors
KW - Sparsity
UR - http://www.scopus.com/inward/record.url?scp=84964293349&partnerID=8YFLogxK
U2 - https://doi.org/10.1214/16-EJS1130
DO - https://doi.org/10.1214/16-EJS1130
M3 - Article
SN - 1935-7524
VL - 10
SP - 976
EP - 1000
JO - Electronic Journal of Statistics
JF - Electronic Journal of Statistics
IS - 1
ER -