A Bayesian nonparametric approach to log-concave density estimation

Bernoulli, vol. 26

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 further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.

Authors 2

1. Ester Mariucci

   University of Potsdam

   Affiliation as printed

   Institut für Mathematik , Universität Potsdam , Karl-Liebknecht-Str. 24-25 , Potsdam 14476 , Germany

2. Botond Szabó Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute , Leiden University , Niels Bohrweg 1 , Leiden 2333 CA , Netherlands