Theoretical Convergence Analysis for Hilbert Space MCMC with Score-based Priors for Nonlinear Bayesian Inverse Problems
Keywords: theory paper, theoretical convergence analysis, nonlinear inverse problems, bayesian inference, hilbert space, Langevin MCMC, score-based generative models
Abstract: In recent years, several works have explored the use of score-based generative models as expressive priors in Markov chain Monte Carlo (MCMC) algorithms for provable posterior sampling, even in the challenging case of nonlinear Bayesian inverse problems. However, these approaches have been mostly limited to finite-dimensional approximations, while the original problems are typically defined in function spaces of infinite dimension. It is well known that algorithms designed for finite-dimensional settings can encounter theoretical and practical issues when applied to infinite-dimensional objects, such as an inconsistent behavior across different discretizations. In this work, we address this limitation by leveraging the recently developed framework for score-based generative models in Hilbert spaces to learn an infinite-dimensional score, which we use as a prior in a function-space Langevin-type MCMC algorithm, providing theoretical guarantees for convergence in the context of nonlinear Bayesian inverse problems. Crucially, we prove that controlling the approximation error of the score is not only essential for ensuring convergence but also that modifying the standard score-based Langevin MCMC through the selection of an appropriate preconditioner is necessary. Our analysis shows how the control over the score approximation error influences the design of the preconditioner---an aspect unique to the infinite-dimensional setting.
Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
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Submission Number: 8131
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