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2014 The horseshoe estimator: Posterior concentration around nearly black vectors
S. L. van der Pas, B. J. K. Kleijn, A. W. van der Vaart
Electron. J. Statist. 8(2): 2585-2618 (2014). DOI: 10.1214/14-EJS962

Abstract

We consider the horseshoe estimator due to Carvalho, Polson and Scott (2010) for the multivariate normal mean model in the situation that the mean vector is sparse in the nearly black sense. We assume the frequentist framework where the data is generated according to a fixed mean vector. We show that if the number of nonzero parameters of the mean vector is known, the horseshoe estimator attains the minimax $\ell_{2}$ risk, possibly up to a multiplicative constant. We provide conditions under which the horseshoe estimator combined with an empirical Bayes estimate of the number of nonzero means still yields the minimax risk. We furthermore prove an upper bound on the rate of contraction of the posterior distribution around the horseshoe estimator, and a lower bound on the posterior variance. These bounds indicate that the posterior distribution of the horseshoe prior may be more informative than that of other one-component priors, including the Lasso.

Citation

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S. L. van der Pas. B. J. K. Kleijn. A. W. van der Vaart. "The horseshoe estimator: Posterior concentration around nearly black vectors." Electron. J. Statist. 8 (2) 2585 - 2618, 2014. https://doi.org/10.1214/14-EJS962

Information

Published: 2014
First available in Project Euclid: 9 December 2014

zbMATH: 1309.62060
MathSciNet: MR3285877
Digital Object Identifier: 10.1214/14-EJS962

Subjects:
Primary: 62F10 , 62F15

Keywords: Bayesian inference , Empirical Bayes , horseshoe prior , normal means model , posterior contraction , Sparsity , worst case risk

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.8 • No. 2 • 2014
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