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Randomized maximum likelihood based posterior sampling

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.

Data Availability

No data are used in the manuscript.

Code Availability

Selected Python codes used in the preparation of this manuscript are available at https://bitbucket.org/JanadeWiljes/workspace/projects/RBPS.

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Funding

Open Access funding provided by NORCE Norwegian Research Centre AS. For this work, Yuming Ba was supported by the China Scholarship Council. Dean Oliver was supported by the NORCE Norwegian Research Centre cooperative research project “Assimilating 4D Seismic Data: Big Data Into Big Models” which is funded by industry partners Aker BP, Equinor, Lundin Norway, Repsol, and Total, as well as the Research Council of Norway through the Petromaks2 program. Jana de Wiljes and Sebastian Reich have been partially funded by Deutsche Forschungsgemeinschaft (DFG) - Project-ID 318763901 - SFB1294.

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Correspondence to Dean S. Oliver.

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Appendix

Appendix

Table 1 Notation used throughout the manuscript

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Ba, Y., de Wiljes, J., Oliver, D.S. et al. Randomized maximum likelihood based posterior sampling. Comput Geosci 26, 217–239 (2022). https://doi.org/10.1007/s10596-021-10100-y

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Keywords

  • Randomized maximum likelihood
  • Importance sampling
  • Minimization
  • Multimodal posterior
  • Bayesian inverse problem