# A Bayesian nonparametric approach to log-concave density estimation

## TL;DR

This paper introduces a Bayesian nonparametric method for estimating log-concave densities that achieves near-minimax convergence rates, with practical computational approaches demonstrated through simulations.

## Contribution

It proposes a novel Bayesian approach using an exponentiated Dirichlet process mixture prior for log-concave density estimation, with theoretical convergence guarantees.

## Key findings

- Posterior converges at near-minimax rate in Hellinger distance.
- Two computationally feasible approximations are developed.
- Empirical Bayes approach is demonstrated through simulations.

## Abstract

The estimation of a log-concave density on $\mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We also present two computationally more feasible approximations and a more practical empirical Bayes approach, which are illustrated numerically via simulations.