Adaptive horseshoe estimation

In our work we consider the horseshoe estimator in the sparse multivariate mean model. We take a frequentist point of view by assuming that the data is generated from a fixed mean vector. In the case the number of nonzero parameters is known it was shown that the horseshoe posterior contracts around the true signal with the minimax rate in \(\ell_2\)-norm. However, in paractice the number of non-zero parameters are unknown. Therefore we consider the empirical Bayes method and show that the empirical Bayes posterior distribution achieves the (nearly) minimax rate. This is an ongoing joint work with Stephanie van der Pas and Aad van der Vaart.