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：我们考虑在科学、工程和成像领域的贝叶斯反问题中从后验分布采样的难题。所提方法属于贝叶斯推断中常用的独立Metropolis-Hastings（IMH）采样算法族。基于存在一个易于采样但可能具有显著偏差的近似后验分布这一前提，我们提出Proximal-IMH方案，该方案通过辅助优化问题对近似后验样本进行校正以消除偏差。这种校正产生一种局部调整机制，在精确模型的保真度与近似参考点的稳定性之间进行权衡。在理想化设定下，我们证明近端校正能缩小近似后验与精确后验之间的差异，从而提升接受率和混合性能。该方法同时适用于线性和非线性输入-输出算子，尤其适用于精确后验采样计算成本过高的反问题。我们开展了包含多峰先验和数据驱动先验与非线性输入-输出算子相结合的数值实验，结果表明Proximal-IMH在可靠性上优于现有IMH变体。