Asymptotic Properties of Bayes Risk for the Horseshoe Prior

Abstract

In this paper, we establish some optimality properties of the multiple testing rule induced by the horseshoe estimator due to Carvalho, Polson, and Scott (2010, 2009) from a Bayesian decision theoretic viewpoint. We consider the two-groups model for the data and an additive loss structure such that the total loss is equal to the number of misclassified hypotheses. We use the same asymptotic framework as Bogdan, Chakrabarti, Frommlet, and Ghosh (2011) who introduced the Bayes oracle in the context of multiple testing and provided conditions under which the Benjamini-Hochberg and Bonferroni procedures attain the risk of the Bayes oracle. We prove a similar result for the horseshoe decision rule up to $O(1)$ with the constant in the horseshoe risk close to the constant in the oracle. We use the Full Bayes estimate of the tuning parameter $\tau$. It is worth noting that the Full Bayes estimate cannot be replaced by the Empirical Bayes estimate, which tends to be too small.