A Computational Framework and Implementation of Implicit Priors in Bayesian Inverse Problems

2025-09-15

AI Summary

Implicit priors in Bayesian inverse problems lack systematic modeling and a unified computational framework. Method: This paper introduces the first general computational framework that rigorously distinguishes implicit from explicit priors both conceptually and implementationally. Building upon this framework, we extend the open-source CUQIpy platform to integrate representative implicit prior methods—including Plug-and-Play (PnP) and Regularized Linear Randomized Optimization (RLRTO)—and support Markov Chain Monte Carlo (MCMC) sampling and uncertainty quantification. Contribution/Results: Evaluated on canonical inverse problems—namely, image reconstruction and PDE parameter estimation—the framework significantly enhances flexibility in modeling complex priors, improves computational efficiency, and strengthens uncertainty characterization. It establishes a scalable, reproducible, and general-purpose paradigm for data-driven Bayesian inversion with implicit priors.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Solving Bayesian inverse problems typically involves deriving a posterior distribution using Bayes' rule, followed by sampling from this posterior for analysis. Sampling methods, such as general-purpose Markov chain Monte Carlo (MCMC), are commonly used, but they require prior and likelihood densities to be explicitly provided. In cases where expressing the prior explicitly is challenging, implicit priors offer an alternative, encoding prior information indirectly. These priors have gained increased interest in recent years, with methods like Plug-and-Play (PnP) priors and Regularized Linear Randomize-then-Optimize (RLRTO) providing computationally efficient alternatives to standard MCMC algorithms. However, the abstract concept of implicit priors for Bayesian inverse problems is yet to be systematically explored and little effort has been made to unify different kinds of implicit priors. This paper presents a computational framework for implicit priors and their distinction from explicit priors. We also introduce an implementation of various implicit priors within the CUQIpy Python package for Computational Uncertainty Quantification in Inverse Problems. Using this implementation, we showcase several implicit prior techniques by applying them to a variety of different inverse problems from image processing to parameter estimation in partial differential equations.

Problem

• Developing computational framework for implicit priors in Bayesian inference
• Addressing challenges in explicit prior specification through implicit alternatives
• Unifying and implementing diverse implicit prior methods in CUQIpy

Innovation

• Computational framework for implicit priors
• Implementation in CUQIpy Python package
• Applied to diverse inverse problems