We consider the problem of sampling from a posterior distribution arising in Bayesian inverse problems in science, engineering, and imaging. Our method belongs to the family of independence Metropolis-Hastings (IMH) sampling algorithms, which are common in Bayesian inference. Relying on the existence of an approximate posterior distribution that is cheaper to sample from but may have significant bias, we introduce Proximal-IMH, a scheme that removes this bias by correcting samples from the approximate posterior through an auxiliary optimization problem. This yields a local adjustment that trades off adherence to the exact model against stability around the approximate reference point. For idealized settings, we prove that the proximal correction tightens the match between approximate and exact posteriors, thereby improving acceptance rates and mixing. The method applies to both linear and nonlinear input-output operators and is particularly suitable for inverse problems where exact posterior sampling is too expensive. We present numerical experiments including multimodal and data-driven priors with nonlinear input-output operators. The results show that Proximal-IMH reliably outperforms existing IMH variants.

翻译：本文研究从贝叶斯反问题中产生的后验分布进行采样的问题，此类问题常见于科学、工程及成像领域。我们的方法属于独立Metropolis-Hastings（IMH）采样算法家族，该族算法在贝叶斯推断中应用广泛。基于存在一种更易采样但可能具有显著偏差的近似后验分布，我们提出了Proximal-IMH方案，该方案通过辅助优化问题校正来自近似后验的样本，从而消除偏差。这产生了一种局部调整机制，在遵循精确模型与保持近似参考点附近的稳定性之间进行权衡。在理想化设定下，我们证明近端校正增强了近似后验与精确后验之间的匹配度，从而提高了接受率与混合效率。该方法适用于线性与非线性输入-输出算子，尤其适用于精确后验采样成本过高的反问题。我们展示了包含多模态先验与数据驱动先验的非线性输入-输出算子数值实验。结果表明，Proximal-IMH在性能上稳定优于现有IMH变体。