  We study the convergence rates of empirical Bayes posterior distributions for
nonparametric and high-dimensional inference. We show that as long as the
hyperparameter set is discrete, the empirical Bayes posterior distribution
induced by the maximum marginal likelihood estimator can be regarded as a
variational approximation to a hierarchical Bayes posterior distribution. This
connection between empirical Bayes and variational Bayes allows us to leverage
the recent results in the variational Bayes literature, and directly obtains
the convergence rates of empirical Bayes posterior distributions from a
variational perspective. For a more general hyperparameter set that is not
necessarily discrete, we introduce a new technique called "prior decomposition"
to deal with prior distributions that can be written as convex combinations of
probability measures whose supports are low-dimensional subspaces. This leads
to generalized versions of the classical "prior mass and testing" conditions
for the convergence rates of empirical Bayes. Our theory is applied to a number
of statistical estimation problems including nonparametric density estimation
and sparse linear regression.
