Adaptive posterior contraction rates for the horseshoe

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

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)3196-3225
Number of pages30
JournalElectronic Journal of Statistics
Volume11
Issue number2
DOIs
Publication statusPublished - 2017
Externally publishedYes

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