Sub-optimality of some continuous shrinkage priors

Two-component mixture priors provide a traditional way to induce sparsity in high-dimensional Bayes models. However, several aspects of such a prior, including computational complexities in high-dimensions, interpretation of exact zeros and non-sparse posterior summaries under standard loss functions, have motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians. Interestingly, we demonstrate that many commonly used shrinkage priors, including the Bayesian Lasso, do not have adequate posterior concentration in high-dimensional settings.