# Empirical priors and posterior concentration rates for a monotone density

## TL;DR

This paper introduces a new empirical Bayes prior for monotone density estimation that achieves near-optimal posterior concentration rates and demonstrates practical advantages over traditional methods like Dirichlet process mixtures.

## Contribution

The paper develops a novel empirical Bayes prior tailored for monotone densities, ensuring desirable concentration properties and near-minimax convergence rates.

## Key findings

- Empirical Bayes prior achieves near-minimax posterior concentration rates.
- Numerical results show improved performance over Dirichlet process mixtures.
- Proposed method satisfies empirical prior concentration conditions.

## 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.