In this article, we investigate the problem of estimating a spatially inhomogeneous function and its derivatives in the white noise model using Besov-Laplace priors. We show that smoothness-matching priors attains minimax optimal posterior contraction rates, in strong Sobolev metrics, over the Besov spaces $B^\beta_{11}$, $\beta > d/2$, closing a gap in the existing literature. Our strong posterior contraction rates also imply that the posterior distributions arising from Besov-Laplace priors with matching regularity enjoy a desirable plug-in property for derivative estimation, entailing that the push-forward measures under differential operators optimally recover the derivatives of the unknown regression function. The proof of our results relies on the novel approach to posterior contraction rates, based on Wasserstein distance, recently developed by Dolera, Favaro and Mainini (Probability Theory and Related Fields, 2024). We show how this approach allows to overcome some technical challenges that emerge in the frequentist analysis of smoothness-matching Besov-Laplace priors.

翻译：本文研究在白噪声模型中使用Besov-Laplace先验估计空间非齐次函数及其导数的问题。我们证明，在Besov空间$B^\beta_{11}$（$\beta > d/2$）上，光滑度匹配先验在强Sobolev度量下达到了极小极大最优后验收缩率，从而填补了现有文献的空白。我们的强后验收缩率结果还表明，具有匹配正则性的Besov-Laplace先验产生的后验分布对于导数估计具有理想的插件性质，这意味着微分算子下的推前测度能够最优地恢复未知回归函数的导数。结果的证明基于Dolera、Favaro和Mainini（《概率论及相关领域》，2024年）最新发展的基于Wasserstein距离的后验收缩率新方法。我们展示了该方法如何克服光滑度匹配Besov-Laplace先验在频率派分析中出现的一些技术挑战。