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 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.

翻译：Besov先验是一类对空间非均匀函数建模的非参数先验，广泛应用于反问题与成像领域，具有促进稀疏性和保持边缘等优异特性。近期一系列工作开始探讨其渐近频率学派收敛性质。本文考虑在密度估计模型中，假设观测数据来自可能具有空间非均匀性的真实密度（属于Besov空间）时，与Besov-Laplace先验相关的后验分布的理论恢复性能。我们改进了现有结果，并证明经过精细调节的Besov-Laplace先验能够实现最优后验收缩速率。进一步地，我们展示了对正则性参数引入超先验的分层过程能够适应任意光滑度水平。