Bayesian linear inverse problems aim to recover an unknown signal from noisy observations, incorporating prior knowledge. This paper analyses a data-dependent method to choose the scale parameter of a Gaussian prior. The method we study arises from early stopping methods, which have been successfully applied to a range of problems, such as statistical inverse problems, in the frequentist setting. These results are extended to the Bayesian setting. We study the use of a discrepancy-based stopping rule in the setting of random noise, which allows for adaptation. Our proposed stopping rule results in optimal rates for the reparameterized problem under certain conditions on the prior covariance operator. We furthermore derive for which class of signals this method is adaptive. It is also shown that the associated posterior contracts at the same rate as the MAP estimator and provides a conservative measure of uncertainty. We implement the proposed stopping rule using the continuous-time ensemble Kalman--Bucy filter (EnKBF). The fictitious time parameter replaces the scale parameter, and the ensemble size is appropriately adjusted in order not to lose the statistical optimality of the computed estimator. With this Monte Carlo algorithm, we extend our results numerically to a nonlinear problem.

翻译：贝叶斯线性反问题旨在结合先验知识，从含噪声观测中恢复未知信号。本文分析了一种基于数据的高斯先验尺度参数选择方法。该方法源于提前终止策略，该策略已在频率学派框架下成功应用于统计反问题等一系列问题。我们将这些结果扩展至贝叶斯框架。我们研究了随机噪声背景下基于残差的终止准则，该方法具备自适应特性。在先验协方差算子满足特定条件时，所提出的终止准则可使重参数化问题达到最优收敛速率。我们进一步推导了该方法适用的信号类别。研究还表明，对应的后验分布以与最大后验概率估计器相同的速率收缩，并能提供保守的不确定性度量。我们采用连续时间集成卡尔曼-布西滤波器实现所提出的终止准则。虚拟时间参数替代了尺度参数，并通过适当调整集成规模以保证计算估计量的统计最优性。借助该蒙特卡洛算法，我们将数值结果扩展至非线性问题。