This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is well known to be thorny, and a recent paper proposes a notably simple and effective solution. Consequently, it offers an remarkable additional flexibility when it comes to estimating observation parameters. The proposed strategy enables us to define an optimal estimator for both the observation parameters and the image of interest. Furthermore, the strategy provides a means of quantifying uncertainty. In addition, MCMC algorithms allow for the efficient computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainties quantification.

翻译：本文探讨了逆问题中观测参数（响应参数与误差参数）的估计问题，重点研究在贝叶斯框架中引入正则化且先验通过扩散过程建模的情形。在此背景下，后验采样问题的复杂性已广为人知，而近期一篇论文提出了一种显著简洁高效的解决方案。因此，该方法为观测参数估计提供了非凡的额外灵活性。所提出的策略使我们能够为观测参数与目标图像定义最优估计量，同时提供了量化不确定性的手段。此外，MCMC算法能够高效计算后验分布的估计值及其性质，并具备一定的理论保证。本文通过多组数值实验，明确验证了所提方法在计算效率、估计质量及不确定性量化方面的优越性。