Combining PushForward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems
We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data on quantities of interest. The solution, given as a probability measure, is derived using a Bayesian updating approach for measurable maps that finds a posterior probability measure, that when propagated through the deterministic model produces a pushforward measure that exactly matches the observed probability measure on the data. Our approach for finding such posterior measures, which we call consistent Bayesian inference or pushforward based inference, is simple and only requires the computation of the pushforward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observationconsistent posteriors and present both stability and error analyses. We also discuss the relationships between consistent Bayesian inference, classical/statistical Bayesian inference, and a recently developed measuretheoretic approach for inference. Finally, analytical and numerical results are presented to highlight certain properties of the consistent Bayesian approach and the differences between this approach and the two aforementioned alternatives for inference.
 Authors:

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 Univ. of Colorado, Boulder, CO (United States)
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Publication Date:
 Report Number(s):
 SAND20173323J
Journal ID: ISSN 10648275; 652120
 Grant/Contract Number:
 AC0494AL85000
 Type:
 Accepted Manuscript
 Journal Name:
 SIAM Journal on Scientific Computing
 Additional Journal Information:
 Journal Volume: 40; Journal Issue: 2; Journal ID: ISSN 10648275
 Publisher:
 SIAM
 Research Org:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org:
 USDOE National Nuclear Security Administration (NNSA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
 OSTI Identifier:
 1469654
Butler, T., Jakeman, J., and Wildey, T.. Combining PushForward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems. United States: N. p.,
Web. doi:10.1137/16m1087229.
Butler, T., Jakeman, J., & Wildey, T.. Combining PushForward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems. United States. doi:10.1137/16m1087229.
Butler, T., Jakeman, J., and Wildey, T.. 2018.
"Combining PushForward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems". United States.
doi:10.1137/16m1087229. https://www.osti.gov/servlets/purl/1469654.
@article{osti_1469654,
title = {Combining PushForward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems},
author = {Butler, T. and Jakeman, J. and Wildey, T.},
abstractNote = {We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data on quantities of interest. The solution, given as a probability measure, is derived using a Bayesian updating approach for measurable maps that finds a posterior probability measure, that when propagated through the deterministic model produces a pushforward measure that exactly matches the observed probability measure on the data. Our approach for finding such posterior measures, which we call consistent Bayesian inference or pushforward based inference, is simple and only requires the computation of the pushforward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observationconsistent posteriors and present both stability and error analyses. We also discuss the relationships between consistent Bayesian inference, classical/statistical Bayesian inference, and a recently developed measuretheoretic approach for inference. Finally, analytical and numerical results are presented to highlight certain properties of the consistent Bayesian approach and the differences between this approach and the two aforementioned alternatives for inference.},
doi = {10.1137/16m1087229},
journal = {SIAM Journal on Scientific Computing},
number = 2,
volume = 40,
place = {United States},
year = {2018},
month = {4}
}