Open Access
May 2020 A Bayesian nonparametric approach to log-concave density estimation
Ester Mariucci, Kolyan Ray, Botond Szabó
Bernoulli 26(2): 1070-1097 (May 2020). DOI: 10.3150/19-BEJ1139


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.


Download Citation

Ester Mariucci. Kolyan Ray. Botond Szabó. "A Bayesian nonparametric approach to log-concave density estimation." Bernoulli 26 (2) 1070 - 1097, May 2020.


Received: 1 March 2017; Revised: 1 June 2019; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166557
MathSciNet: MR4058361
Digital Object Identifier: 10.3150/19-BEJ1139

Keywords: convergence rate , Density estimation , Dirichlet mixture , Log-concavity , nonparametric hypothesis testing , posterior distribution

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 2 • May 2020
Back to Top