Bayesian model inversion using stochastic spectral embedding

In this paper we propose a new sampling-free approach to solve Bayesian model inversion problems that is an extension of the previously proposed spectral likelihood expansions (SLE) method. Our approach, called stochastic spectral likelihood embedding (SSLE), uses the recently presented stochastic spectral embedding (SSE) method for local spectral expansion refinement to approximate the likelihood function at the core of Bayesian inversion problems. We show that, similar to SLE, this approach results in analytical expressions for key statistics of the Bayesian posterior distribution, such as evidence, posterior moments and posterior marginals, by direct post-processing of the expansion coefficients. Because SSLE and SSE rely on the direct approximation of the likelihood function, they are in a way independent of the computational/mathematical complexity of the forward model. We further enhance the efficiency of SSLE by introducing a likelihood specific adaptive sample enrichment scheme. To showcase the performance of the proposed SSLE, we solve three problems that exhibit different kinds of complexity in the likelihood function: multimodality, high posterior concentration and high nominal dimensionality. We demonstrate how SSLE significantly improves on SLE, and present it as a promising alternative to existing inversion frameworks. (C) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

In this paper, we propose a new Bayesian model inversion approach called SSLE, which approximates the likelihood function to obtain key statistics of the posterior distribution without being affected by the complexity of the forward model. The efficiency of SSLE is further enhanced by an adaptive sample enrichment scheme, making it suitable for a variety of challenging likelihood function problems.