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.

翻译：Besov先验是一种能够建模空间非齐次函数的非参数先验，常用于反问题和图像处理领域，展现出吸引人的稀疏促进和边缘保持特性。近期一系列研究开启了对其渐近频率学派收敛性质的探讨。本文考虑密度估计模型中与Besov-Laplace先验相关联的后验分布的理论恢复性能，假设观测数据由可能属于Besov空间的空间非齐次真实密度生成。我们改进了现有结果，并证明经过精细调整的Besov-Laplace先验能够达到最优后验收缩率。此外，我们表明，涉及正则化参数超先验的分层方法可实现对任意光滑水平的自适应。