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

We are considering the problem of sampling from a posterior distribution related to Bayesian inverse problems arising in science, engineering, and imaging. Our method belongs to the family of independence Metropolis--Hastings (IMH) sampling algorithms. These are quite common in Bayesian inference. Relying on the existence of an approximate posterior distribution that is cheaper to sample from but can have significant bias, we introduce Proximal-IMH, a scheme that removes this bias: it corrects samples from the approximate posterior solving an auxiliary optimization problem, yielding 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, and thereby improves acceptance rates and mixing. The new method works with both linear and nonlinear input-output operators and is especially suitable for inverse problems where exact posterior sampling is too expensive. We perform several numerical experiments that include multimodal and data-driven priors and nonlinear input-output operators. The results show that Proximal-IMH reliably outperforms existing IMH variants.