Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operators

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.