Title: Empirical priors and posterior concentration rates for a monotone density,
Abstract: In a Bayesian context, prior specification for inference on monotone
densities is conceptually straightforward, but proving posterior convergence
theorems is complicated by the fact that desirable prior concentration
properties often are not satisfied. In this paper, I first develop a new prior
designed specifically to satisfy an empirical version of the prior
concentration property, and then I give sufficient conditions on the prior
inputs such that the corresponding empirical Bayes posterior concentrates
around the true monotone density at nearly the optimal minimax rate. Numerical
illustrations also reveal the practical benefits of the proposed empirical
Bayes approach compared to Dirichlet process mixtures.