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We study the convergence rates of empirical Bayes posterior distributions fornonparametric and high-dimensional inference. We show that as long as thehyperparameter set is discrete, the empirical Bayes posterior distributioninduced by the maximum marginal likelihood estimator can be regarded as avariational approximation to a hierarchical Bayes posterior distribution. Thisconnection between empirical Bayes and variational Bayes allows us to leveragethe recent results in the variational Bayes literature, and directly obtainsthe convergence rates of empirical Bayes posterior distributions from avariational perspective. For a more general hyperparameter set that is notnecessarily discrete, we introduce a new technique called "prior decomposition"to deal with prior distributions that can be written as convex combinations ofprobability measures whose supports are low-dimensional subspaces. This leadsto generalized versions of the classical "prior mass and testing" conditionsfor the convergence rates of empirical Bayes. Our theory is applied to a numberof statistical estimation problems including nonparametric density estimationand sparse linear regression.

Convergence Rates of Empirical Bayes Posterior Distributions: A Variational Perspective