In many scientific applications the aim is to infer a function which is smooth in some areas, but rough or even discontinuous in other areas of its domain. Such spatially inhomogeneous functions can be modelled in Besov spaces with suitable integrability parameters. In this work we study adaptive Bayesian inference over Besov spaces, in the white noise model from the point of view of rates of contraction, using $p$-exponential priors, which range between Laplace and Gaussian and possess regularity and scaling hyper-parameters. To achieve adaptation, we employ empirical and hierarchical Bayes approaches for tuning these hyper-parameters. Our results show that, while it is known that Gaussian priors can attain the minimax rate only in Besov spaces of spatially homogeneous functions, Laplace priors attain the minimax or nearly the minimax rate in both Besov spaces of spatially homogeneous functions and Besov spaces permitting spatial inhomogeneities.

翻译：在许多科学应用中，目标是推断一个在其定义域某些区域光滑、但在其他区域粗糙甚至不连续的函数。此类空间非均匀函数可通过具有适当可积性参数的Besov空间建模。本研究从收缩速率角度出发，在白色噪声模型中研究基于$p$-指数先验（介于拉普拉斯先验与高斯先验之间，且具有正则性与尺度超参数）的Besov空间自适应贝叶斯推断。为实现自适应，我们采用经验贝叶斯和层次贝叶斯方法调节这些超参数。结果表明：尽管已知高斯先验仅在空间均匀函数的Besov空间中能达到极小极大最优速率，但拉普拉斯先验在空间均匀函数和允许空间非均匀性的Besov空间中均能达到或接近极小极大最优速率。