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

Independent Metropolis–Hastings (IMH) algorithms are widely used in Bayesian inference, but their efficiency deteriorates when proposal distributions are constructed from inaccurate or approximate models. We introduce Proximal-IMH, an IMH method that enhances proposal distributions through a proximal posterior correction. Given an approximate posterior sample, each proposal is generated by minimizing a quadratically regularized surrogate objective, producing a local correction that balances fidelity to the exact model with stability around the approximate state. We analyze the resulting proposals from an optimization and probabilistic perspective, showing how the proximal correction improves alignment between approximate and exact posteriors and leads to improved acceptance and mixing behavior. The proposed framework applies to both linear and nonlinear forward operators, and is particularly well suited to Bayesian inverse problems where exact posterior sampling is computationally prohibitive. Numerical experiments on inverse problems with approximate forward models, including nonlinear operators, demonstrate that Proximal-IMH consistently outperforms existing IMH variants while retaining their simplicity and scalability.