Electronic Journal of Statistics

Adaptive posterior contraction rates for the horseshoe

Stéphanie van der Pas, Botond Szabó, and Aad van der Vaart

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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.

Article information

Electron. J. Statist. Volume 11, Number 2 (2017), 3196-3225.

Received: February 2017
First available in Project Euclid: 22 September 2017

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Digital Object Identifier

Primary: 62G15: Tolerance and confidence regions
Secondary: 62F15: Bayesian inference

Horseshoe sparsity nearly black vectors normal means problem adaptive inference frequentist Bayes

Creative Commons Attribution 4.0 International License.


van der Pas, Stéphanie; Szabó, Botond; van der Vaart, Aad. Adaptive posterior contraction rates for the horseshoe. Electron. J. Statist. 11 (2017), no. 2, 3196--3225. doi:10.1214/17-EJS1316. https://projecteuclid.org/euclid.ejs/1506067214

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