Tail-adaptive Bayesian shrinkage

Modern genomic studies are increasingly focused on discovering more and more interesting genes associated with a health response. Traditional shrinkage priors are primarily designed to detect a handful of signals from tens of thousands of predictors in the so-called ultra-sparsity domain. However, they may fail to identify signals when the degree of sparsity is moderate. Robust sparse estimation under diverse sparsity regimes relies on a tail-adaptive shrinkage property. In this property, the tail-heaviness of the prior adjusts adaptively, becoming larger or smaller as the sparsity level increases or decreases, respectively, to accommodate more or fewer signals. In this study, we propose a global-local-tail (GLT) Gaussian mixture distribution that ensures this property. We examine the role of the tail-index of the prior in relation to the underlying sparsity level and demonstrate that the GLT posterior contracts at the minimax optimal rate for sparse normal mean models. We apply both the GLT prior and the Horseshoe prior to real data problems and simulation examples. Our findings indicate that the varying tail rule based on the GLT prior offers advantages over a fixed tail rule based on the Horseshoe prior in diverse sparsity regimes.