Uncertainty quantification for the horseshoe (with discussion). (English) Zbl 1384.62155

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

Bayesian Anal. 12, No. 4, 1221-1274 (2017).

Summary: We investigate the credible sets and marginal credible intervals resulting from the horseshoe prior in the sparse multivariate normal means model. We do so in an adaptive setting without assuming knowledge of the sparsity level (number of signals). We consider both the hierarchical Bayes method of putting a prior on the unknown sparsity level and the empirical Bayes method with the sparsity level estimated by maximum marginal likelihood. We show that credible balls and marginal credible intervals have good frequentist coverage and optimal size if the sparsity level of the prior is set correctly. By general theory honest confidence sets cannot adapt in size to an unknown sparsity level. Accordingly the hierarchical and empirical Bayes credible sets based on the horseshoe prior are not honest over the full parameter space. We show that this is due to over-shrinkage for certain parameters and characterise the set of parameters for which credible balls and marginal credible intervals do give correct uncertainty quantification. In particular we show that the fraction of false discoveries by the marginal Bayesian procedure is controlled by a correct choice of cut-off.

MSC:

62G15	Nonparametric tolerance and confidence regions
62F15	Bayesian inference
62H12	Estimation in multivariate analysis

Keywords:

credible sets; horseshoe; sparsity; nearly black vectors; normal means problem; frequentist Bayes