Prior normalization for certified likelihood-informed subspace detection of Bayesian inverse problems

TL;DR

This paper introduces a prior normalization technique that transforms non-Gaussian priors into Gaussian ones, enabling the use of likelihood-informed subspace methods to improve MCMC sampling efficiency in high-dimensional Bayesian inverse problems.

Contribution

It proposes a novel prior normalization approach for non-Gaussian priors, facilitating LIS-based MCMC acceleration in high-dimensional settings.

Findings

Effective transformation of heavy-tailed priors into Gaussian form.

Enhanced MCMC sampling efficiency demonstrated on nonlinear inverse problems.

Theoretical analysis confirms integration of normalization with various MCMC methods.

Abstract

Markov chain Monte Carlo (MCMC) methods form one of the algorithmic foundations of Bayesian inverse problems. The recent development of likelihood-informed subspace (LIS) methods offers a viable route to designing efficient MCMC methods for exploring high-dimensional posterior distributions via exploiting the intrinsic low-dimensional structure of the underlying inverse problem. However, existing LIS methods and the associated performance analysis often assume that the prior distribution is Gaussian. This assumption is limited for inverse problems aiming to promote sparsity in the parameter estimation, as heavy-tailed priors, e.g., Laplace distribution or the elastic net commonly used in Bayesian LASSO, are often needed in this case. To overcome this limitation, we consider a prior normalization technique that transforms any non-Gaussian (e.g. heavy-tailed) priors into standard Gaussian distributions, which makes it possible to implement LIS methods to accelerate MCMC sampling via such transformations. We also rigorously investigate the integration of such transformations with several MCMC methods for high-dimensional problems. Finally, we demonstrate various aspects of our theoretical claims on two nonlinear inverse problems.