Abstract
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.
Highlights
Computational models are an invaluable tool for decision making, scientific advances and engineering breakthroughs
While such approaches can often be used in practical applications, they tend not to provide measures of the uncertainties associated with the inferred model input or predictions
It is clear that both stochastic spectral embedding (SSE) algorithms outperform spectral likelihood expansions (SLE), while the adSSE approach manages to improve the performance of SLE by an order of magnitude
Summary
Computational models are an invaluable tool for decision making, scientific advances and engineering breakthroughs. Blatman and Sudret (2011)), its computation has become feasible even in the presence of complex and computationally expensive engineering models This technique has been successfully used in conjunction with MCMC to reduce the total computational costs associated with sampling from the posterior distribution (Marzouk et al, 2007; Wagner et al, 2020). Nagel and Sudret (2016) proposed spectral likelihood expansions (SLE), a novel approach that is closely related to PCE and aims at finding a spectral expansion of the likelihood function in an orthogonal basis w.r.t. the prior density function The advantage of this method is that it provides analytical expressions for the posterior marginals and general posterior moments by post-processing the expansion coefficients.
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