We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise. We consider a class of Gaussian prior probability measures with covariance operator indexed by a hyperparameter that quantifies regularity. By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter. In addition, we show that this adaptive Bayes procedure provides optimal contraction rates up to a slowly varying term and an arbitrarily small constant, without knowledge about the regularity index. Our method needs not the prior covariance, noise covariance and forward operator have a common basis in their singular value decomposition, enlarging the application range compared with the existing results. A simple simulation example is given that illustrates the effectiveness of the proposed method.

中文翻译：

---

非对角假设下经验贝叶斯方法求解反问题的后收缩

我们研究了经验贝叶斯非参数方法，该方法解决了具有高斯先验和高斯噪声的线性反问题的问题。我们考虑一类高斯先验概率测度，其协方差算子由量化正则性的超参数索引。通过引入两个辅助问题，我们构造了经验贝叶斯方法，并证明该方法可以自动选择超参数。此外，我们证明了这种自适应贝叶斯程序可以提供最佳的收缩率，直到一个缓慢变化的项和一个任意小的常数，而无需了解规则指数。我们的方法不需要先验协方差，噪声协方差和前向算子在其奇异值分解中具有共同的基础，与现有结果相比，扩大了应用范围。