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

Besov priors are nonparametric priors that 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 the asymptotic frequentist convergence properties of Besov priors. In the present paper, we consider the theoretical recovery performance of the associated posterior distributions in the density estimation model, under the assumption that the observations are generated by a spatially inhomogeneous true density belonging to a Besov space. We improve on existing results and show that carefully tuned Besov priors attain optimal posterior contraction rates. Furthermore, we show that a hierarchical procedure involving a hyper-prior on the regularity parameter leads to adaptation to any smoothness level.