  In high-dimensional problems, choosing a prior distribution such that the
corresponding posterior has desirable practical and theoretical properties can
be challenging. This begs the question: can the data be used to help choose a
good prior? In this paper, we develop a general strategy for constructing a
data-driven or empirical prior and sufficient conditions for the corresponding
posterior distribution to achieve a certain concentration rate. The idea is
that the prior should put sufficient mass on parameter values for which the
likelihood is large. An interesting byproduct of this data-driven centering is
that the asymptotic properties of the posterior are less sensitive to the prior
shape which, in turn, allows users to work with priors of computationally
convenient forms while maintaining the desired rates. General results on both
adaptive and non-adaptive rates based on empirical priors are presented, along
with illustrations in density estimation, nonparametric regression, and
high-dimensional structured normal models.
