Proximal-IMH: 用于独立Metropolis-Hastings的近端后验提议
Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operators
AI总结：
本文提出了一种改进的独立Metropolis-Hastings算法，通过引入辅助优化问题来消除近似后验分布中的偏差，从而在保持精确模型的同时提高稳定性和采样效率。
摘要：
我们考虑了在科学、工程和成像中的贝叶斯反问题中从后验分布采样的问题。我们的方法属于独立Metropolis-Hastings（IMH）采样算法家族，常用于贝叶斯推断。依赖于存在一个更便宜但可能有显著偏差的近似后验分布，我们引入了Proximal-IMH，通过辅助优化问题纠正近似后验的样本，从而在精确模型和近似参考点周围获得局部调整。对于理想化设置，我们证明了近端校正能够收紧近似和精确后验之间的匹配，从而提高接受率和混合性。该方法适用于线性和非线性输入-输出算子，并特别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和数据驱动先验的数值实验，结果表明Proximal-IMH在现有IMH变体中表现更优。
英文摘要：
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.
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