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Large-scale modern data often involves estimation and testing forhigh-dimensional unknown parameters. It is desirable to identify the sparsesignals, ``the needles in the haystack'', with accuracy and false discoverycontrol. However, the unprecedented complexity and heterogeneity in modern datastructure require new machine learning tools to effectively exploitcommonalities and to robustly adjust for both sparsity and heterogeneity. Inaddition, estimates for high-dimensional parameters often lack uncertaintyquantification. In this paper, we propose a novel Spike-and-Nonparametricmixture prior (SNP) -- a spike to promote the sparsity and a nonparametricstructure to capture signals. In contrast to the state-of-the-art methods, theproposed methods solve the estimation and testing problem at once with severalmerits: 1) an accurate sparsity estimation; 2) point estimates withshrinkage/soft-thresholding property; 3) credible intervals for uncertaintyquantification; 4) an optimal multiple testing procedure that controls falsediscovery rate. Our method exhibits promising empirical performance on bothsimulated data and a gene expression case study.

Nonparametric Empirical Bayes Estimation and Testing for Sparse and Heteroscedastic Signals