A Bayesian nonparametric approach to log-concave density estimation

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.

39 pages, 17 figures. Simulation studies were significantly expanded and one more theorem has been added

References in corpus (3)

• Rates of contraction of posterior distributions based on Gaussian process priors
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Inference and Modeling with Log-concave Distributions

Cited by in corpus (2)

• Nonparametric Bayesian inference for reversible multi-dimensional diffusions
• Nonparametric Bayesian estimation in a multidimensional diffusion model with high frequency data