Abstract
Minimization of a stochastic cost function is commonly used for approximate sampling in high-dimensional Bayesian inverse problems with Gaussian prior distributions and multimodal posterior distributions. The density of the samples generated by minimization is not the desired target density, unless the observation operator is linear, but the distribution of samples is useful as a proposal density for importance sampling or for Markov chain Monte Carlo methods. In this paper, we focus on applications to sampling from multimodal posterior distributions in high dimensions. We first show that sampling from multimodal distributions is improved by computing all critical points instead of only minimizers of the objective function. For applications to high-dimensional geoscience inverse problems, we demonstrate an efficient approximate weighting that uses a low-rank Gauss-Newton approximation of the determinant of the Jacobian. The method is applied to two toy problems with known posterior distributions and a Darcy flow problem with multiple modes in the posterior.
Highlights
In several fields, including groundwater management, groundwater remediation, and petroleum reservoir management, there is a need to characterize permeable rock bodiesthe posterior probability density for reservoir properties, conditional to rate and pressure observations, is typically complex and not sampled
Unlike previous methods that compute minimizers only, we show that exact sampling is possible when all critical points of a stochastic objective function are computed and properly weighted
For all critical points in high dimensions does not appear to be feasible, but we demonstrate that GaussNewton approximations of the weights provide good approximations for minimizers of the objective function in problems with multimodal posteriors
Summary
Importance sampling methods can be considered to be exact sampling methods as they implement Bayes rule directly They are very difficult to apply in high dimensions, as an efficient implementation requires a proposal density that is a good approximation of the posterior [5, 28]. The distribution of samples based on minimizers of the objective function was shown to be correct for Gauss-linear problems, but when the observation operator g(m) is nonlinear, it was necessary to weight the samples because the sampling was only approximate in that case. For all critical points in high dimensions does not appear to be feasible, but we demonstrate that GaussNewton approximations of the weights provide good approximations for minimizers of the objective function in problems with multimodal posteriors.
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