Besov-Laplace priors in density estimation: optimal posterior contraction rates and adaptation

TL;DR

This paper studies the theoretical performance of Besov-Laplace priors in density estimation, demonstrating they achieve optimal convergence rates and adapt to various smoothness levels through hierarchical procedures.

Contribution

It provides new theoretical results showing Besov-Laplace priors attain optimal posterior contraction rates and adaptivity in density estimation.

Findings

Besov-Laplace priors achieve optimal contraction rates.

Hierarchical hyper-priors enable adaptation to unknown smoothness.

Improved theoretical understanding of nonparametric Bayesian density estimation.

Abstract

Besov priors are nonparametric priors that can model spatially inhomogeneous functions. They are routinely used in inverse problems and imaging, where they exhibit attractive sparsity-promoting and edge-preserving features. A recent line of work has initiated the study of their asymptotic frequentist convergence properties. In the present paper, we consider the theoretical recovery performance of the posterior distributions associated to Besov-Laplace priors in the density estimation model, under the assumption that the observations are generated by a possibly spatially inhomogeneous true density belonging to a Besov space. We improve on existing results and show that carefully tuned Besov-Laplace priors attain optimal posterior contraction rates. Furthermore, we show that hierarchical procedures involving a hyper-prior on the regularity parameter lead to adaptation to any smoothness level.