Exact Optimality of the Horseshoe+ Prior

Music City Center 

This talk considers the exact optimality of the horseshoe+ prior in estimation and multiple testing of multivariate normal means under sparsity. First, the posterior means under the horseshoe+ prior are shown to be minimax as point
estimates of the multivariate normal means under sparsity. Then, under the framework of Bogdan et al. (2011), a thresholding multiple testing procedure using the horseshoe+ prior is shown to attain asymptotic Bayes optimality under sparsity (ABOS) property. As the choice of the global parameter is based on the unknown sparsity level, we further propose an empirical Bayes approach for the multiple testing problem and demonstrate its optimality.
The paper also includes some discussions on its connection to the work of van der Pas et al. (2016) and Bhadra et al. (2017).