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.

Many scientific and engineering problems involve estimating hidden quantities from indirect and noisy measurements. Examples include reconstructing medical images, identifying material properties, or estimating an unknown object from scattered waves. Bayesian methods are useful because they provide not only one estimate, but also uncertainty about the answer. However, accurately sampling from these Bayesian distributions can be very expensive when each sample requires solving a large physical model. This paper introduces Proximal-IMH, a faster sampling method for such problems. The method first draws samples using a cheaper approximate model, then corrects each sample through a small optimization problem so that it better matches the accurate model. This correction reduces the bias introduced by the approximate model while keeping the computation efficient. We show theoretically that this correction can improve sampling quality and convergence. In experiments on imaging and wave-based inverse problems, Proximal-IMH gives more accurate samples and converges faster than existing approximate sampling methods.