Continuous Gaussian mixture solution for linear Bayesian inversion with application to Laplace priors

TL;DR

This paper develops a Gaussian mixture-based approach for Bayesian inverse problems with Gaussian likelihoods and Laplace priors, introducing a dimension-reduction technique for efficient sampling of the posterior distribution.

Contribution

It derives a closed-form expression for the posterior Gaussian mixture and proposes a dimension-reduction method to approximate the posterior mixing density in high dimensions.

Findings

Samples from the approximate posterior have low correlation.

The approximation closely matches the exact posterior.

The method is effective for Laplace priors in high-dimensional settings.

Abstract

We focus on Bayesian inverse problems with Gaussian likelihood, linear forward model, and priors that can be formulated as a Gaussian mixture. Such a mixture is expressed as an integral of Gaussian density functions weighted by a mixing density over the mixing variables. Within this framework, the corresponding posterior distribution also takes the form of a Gaussian mixture, and we derive the closed-form expression for its posterior mixing density. To sample from the posterior Gaussian mixture, we propose a two-step sampling method. First, we sample the mixture variables from the posterior mixing density, and then we sample the variables of interest from Gaussian densities conditioned on the sampled mixing variables. However, the posterior mixing density is relatively difficult to sample from, especially in high dimensions. Therefore, we propose to replace the posterior mixing density by a dimension-reduced approximation, and we provide a bound in the Hellinger distance for the resulting approximate posterior. We apply the proposed approach to a posterior with Laplace prior, where we introduce two dimension-reduced approximations for the posterior mixing density. Our numerical experiments indicate that samples generated via the proposed approximations have very low correlation and are close to the exact posterior.