Open Access
2017 Adaptive posterior contraction rates for the horseshoe
Stéphanie van der Pas, Botond Szabó, Aad van der Vaart
Electron. J. Statist. 11(2): 3196-3225 (2017). DOI: 10.1214/17-EJS1316


We investigate the frequentist properties of Bayesian procedures for estimation based on the horseshoe prior in the sparse multivariate normal means model. Previous theoretical results assumed that the sparsity level, that is, the number of signals, was known. We drop this assumption and characterize the behavior of the maximum marginal likelihood estimator (MMLE) of a key parameter of the horseshoe prior. We prove that the MMLE is an effective estimator of the sparsity level, in the sense that it leads to (near) minimax optimal estimation of the underlying mean vector generating the data. Besides this empirical Bayes procedure, we consider the hierarchical Bayes method of putting a prior on the unknown sparsity level as well. We show that both Bayesian techniques lead to rate-adaptive optimal posterior contraction, which implies that the horseshoe posterior is a good candidate for generating rate-adaptive credible sets.


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Stéphanie van der Pas. Botond Szabó. Aad van der Vaart. "Adaptive posterior contraction rates for the horseshoe." Electron. J. Statist. 11 (2) 3196 - 3225, 2017.


Received: 1 February 2017; Published: 2017
First available in Project Euclid: 22 September 2017

zbMATH: 1373.62140
MathSciNet: MR3705450
Digital Object Identifier: 10.1214/17-EJS1316

Primary: 62G15
Secondary: 62F15

Keywords: adaptive inference , frequentist Bayes , horseshoe , nearly black vectors , normal means problem , Sparsity

Vol.11 • No. 2 • 2017
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