A Bayesian nonparametric approach to log-concave density estimation

The estimation of a log-concave density on 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.

Metadaten
Author details:	Ester Mariucci ORCiD, Kolyan Ray ORCiD, Botond Szabo ORCiD
DOI:	https://doi.org/10.3150/19-BEJ1139
ISSN:	1350-7265
ISSN:	1573-9759
Title of parent work (English):	Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
Publisher:	International Statistical Institute
Place of publishing:	The Hague
Publication type:	Article
Language:	English
Date of first publication:	2020/01/31
Publication year:	2020
Release date:	2023/06/09
Tag:	Dirichlet mixture; convergence rate; density estimation; log-concavity; nonparametric hypothesis testing; posterior distribution
Volume:	26
Issue:	2
Number of pages:	28
First page:	1070
Last Page:	1097
Funding institution:	European Research CouncilEuropean Research Council (ERC)European; Commission [320637]; Federal Ministry for Education and Research by; Alexander von Humboldt Foundation; Deutsche Forschungsgemeinschaft (DFG,; German Research Foundation)German Research Foundation (DFG) [314838170,; GRK 2297 MathCoRe]; Deutsche Forschungsgemeinschaft (DFG)German Research; Foundation (DFG) [CRC 1294]; Netherlands Organization for Scientific; Research (NWO)Netherlands Organization for Scientific Research (NWO); [639.031.654]
Organizational units:	Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
DDC classification:	5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Peer review:	Referiert
Publishing method:	Open Access / Green Open-Access