In high and infinite-dimensional problems, the Bayesian prior specification can be a challenge.  For example, in high-dimensional regression, while sparsity considerations drive the choice of prior on the model, there is no genuine prior information available about the coefficients in a given model.  Moreover, the choice of prior for the model-specific parameters impacts both the computational and theoretical performance of the posterior.  As an alternative, one might be tempted to choose a computationally simple "informative" empirical prior on the model-specific parameters, depending on data in a suitable way.  In this talk, I will present a new approach for empirical prior specification in high-dimensional problems, based on the idea of data-driven prior centering.  I will give (adaptive) concentration rate results for this new "empirical Bayes" posterior in several specific examples, with illustrations, and I will also say a few words about the general construction and corresponding theory.