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.
Research partially supported by MUR, PRIN project 2022CLTYP4.
We would like to thank Kolyan Ray, Judith Rousseau and Sergios Agapiou for valuable discussions. We are also grateful to the Associate Editor and to two anonymous Referees for very helpful comments that lead to an improvement of the paper. The first version of the manuscript was completed while M.G. was affiliated with the University of Oxford and supported by the ERC grant agreement No. 834275 (GTBB).
"Besov-Laplace priors in density estimation: optimal posterior contraction rates and adaptation." Electron. J. Statist. 17 (2) 2210 - 2249, 2023. https://doi.org/10.1214/23-EJS2161