Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operators
arXiv:2602.21426v1 Announce Type: new Abstract: 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 particula
arXiv:2602.21426v1 Announce Type: new Abstract: 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.
Executive Summary
This article introduces Proximal-IMH, a novel Metropolis-Hastings sampling algorithm designed to address the limitations of existing independence Metropolis-Hastings (IMH) methods in Bayesian inverse problems. By incorporating a proximal correction, Proximal-IMH effectively removes bias from approximate posterior distributions, enabling improved acceptance rates and mixing. The proposed method is applicable to both linear and nonlinear operators and exhibits robust performance in numerical experiments. The results demonstrate Proximal-IMH's reliability in outperforming existing IMH variants, particularly in scenarios where exact posterior sampling is computationally expensive. As such, Proximal-IMH provides a promising solution for efficient Bayesian inference in complex inverse problems.
Key Points
- ▸ Proximal-IMH addresses the limitations of existing IMH methods in Bayesian inverse problems.
- ▸ The proximal correction removes bias from approximate posterior distributions.
- ▸ Proximal-IMH exhibits improved acceptance rates and mixing compared to existing IMH variants.
Merits
Generalizability
Proximal-IMH is applicable to both linear and nonlinear operators, making it a versatile method for Bayesian inference in various inverse problems.
Efficiency
The proposed method enables efficient sampling from approximate posterior distributions, even in scenarios where exact posterior sampling is computationally expensive.
Robustness
Proximal-IMH demonstrates robust performance in numerical experiments, including multimodal and data-driven priors with nonlinear input-output operators.
Demerits
Computational Complexity
The proximal correction requires solving an auxiliary optimization problem, which may introduce additional computational complexity.
Scalability
The performance of Proximal-IMH in large-scale inverse problems remains unclear and requires further investigation.
Expert Commentary
The introduction of Proximal-IMH represents a significant step forward in the development of efficient Bayesian inference methods for inverse problems. By leveraging the proximal correction, the proposed algorithm effectively addresses the limitations of existing IMH methods and demonstrates robust performance in numerical experiments. However, the computational complexity and scalability of Proximal-IMH require further investigation to ensure its applicability in large-scale inverse problems. Nevertheless, the potential of Proximal-IMH to facilitate efficient and accurate Bayesian inference in complex systems makes it an attractive solution for various scientific and engineering applications.
Recommendations
- ✓ Future research should focus on investigating the performance of Proximal-IMH in large-scale inverse problems and exploring potential combinations with other Bayesian inference methods, such as HMC.
- ✓ The development of Proximal-IMH highlights the need for further research on efficient Bayesian inference methods, particularly in scenarios where exact posterior sampling is computationally expensive.
