  We consider the asymptotic behaviour of the marginal maximum likelihood
empirical Bayes posterior distribution in general setting. First we
characterize the set where the maximum marginal likelihood estimator is located
with high probability. Then we provide oracle type of upper and lower bounds
for the contraction rates of the empirical Bayes posterior. We also show that
the hierarchical Bayes posterior achieves the same contraction rate as the
maximum marginal likelihood empirical Bayes posterior. We demonstrate the
applicability of our general results for various models and prior distributions
by deriving upper and lower bounds for the contraction rates of the
corresponding empirical and hierarchical Bayes posterior distributions.
