Sequential Ensemble Transform for Bayesian Inverse Problems

TL;DR

The paper introduces the Sequential Ensemble Transform (SET) method, a novel approach for sampling from Bayesian posteriors that uses optimal transport, demonstrating convergence, robustness, and computational efficiency over traditional methods.

Contribution

The SET method is a new approach that employs optimal transport to generate approximate Bayesian posterior samples, with proven convergence and improved robustness and efficiency.

Findings

SET converges weakly to the true posterior as sample size increases.

SET is more robust to Markov kernel choices than standard SMC methods.

SET requires less computational effort to achieve similar accuracy.

Abstract

We present the Sequential Ensemble Transform (SET) method, an approach for generating approximate samples from a Bayesian posterior distribution. The method explores the posterior distribution by solving a sequence of discrete optimal transport problems to produce a series of transport plans which map prior samples to posterior samples. We prove that the sequence of Dirac mixture distributions produced by the SET method converges weakly to the true posterior as the sample size approaches infinity. Furthermore, our numerical results indicate that, when compared to standard Sequential Monte Carlo (SMC) methods, the SET approach is more robust to the choice of Markov mutation kernels and requires less computational efforts to reach a similar accuracy when used to explore complex posterior distributions. Finally, we describe adaptive schemes that allow to completely automate the use of the SET method.