Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems

Butler, T.; Jakeman, J.; Wildey, T.

doi:10.1137/16m1087229

Title: Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems

Abstract

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 push-forward 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 push-forward based inference, is simple and only requires the computation of the push-forward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observation-consistent 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 measure-theoretic 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:

Butler, T. ^[1]; Jakeman, J. ^[2]; Wildey, T. ^[2]

Univ. of Colorado, Boulder, CO (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Publication Date:: Tue Apr 03 00:00:00 EDT 2018

Research Org.:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1469654

Report Number(s):: SAND2017-3323J
Journal ID: ISSN 1064-8275; 652120

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Accepted Manuscript

Journal Name:: SIAM Journal on Scientific Computing

Additional Journal Information:: Journal Volume: 40; Journal Issue: 2; Journal ID: ISSN 1064-8275

Publisher:: SIAM

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING

Citation Formats


                    Butler, T., Jakeman, J., and Wildey, T. Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems.  United States: N. p., 2018. 
Web.  doi:10.1137/16m1087229.

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                    Butler, T., Jakeman, J., & Wildey, T. Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems.  United States.  https://doi.org/10.1137/16m1087229

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                    Butler, T., Jakeman, J., and Wildey, T. Tue .  
"Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems".  United States.  https://doi.org/10.1137/16m1087229.  https://www.osti.gov/servlets/purl/1469654.

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@article{osti_1469654,

  title        = {Combining Push-Forward 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 push-forward 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 push-forward based inference, is simple and only requires the computation of the push-forward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observation-consistent 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 measure-theoretic 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         = {Tue Apr 03 00:00:00 EDT 2018},

  month        = {Tue Apr 03 00:00:00 EDT 2018}

}

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Works referenced in this record:

A Measure-Theoretic Computational Method for Inverse Sensitivity Problems I: Method and Analysis
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Breidt, J.; Butler, T.; Estep, D.
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Measure-theoretic complexity of ergodic systems
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Fusion of hard and soft information in nonparametric density estimation
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Royset, Johannes O.; Wets, Roger J-B
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Inverse problems: A Bayesian perspective
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The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
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Works referencing / citing this record:

Data-driven uncertainty quantification for predictive flow and transport modeling using support vector machines
journal, August 2018

He, Jiachuan; Mattis, Steven A.; Butler, Troy D.
Computational Geosciences, Vol. 23, Issue 4
DOI: 10.1007/s10596-018-9762-4

Efficient parameter estimation for a methane hydrate model with active subspaces
journal, August 2018

Teixeira Parente, Mario; Mattis, Steven; Gupta, Shubhangi
Computational Geosciences, Vol. 23, Issue 2
DOI: 10.1007/s10596-018-9769-x

Efficient parameter estimation for a methane hydrate model with active subspaces
text, January 2018

Parente, Mario Teixeira; Mattis, Steven; Gupta, Shubhangi
arXiv
DOI: 10.48550/arxiv.1801.09499

Specification of Additional Information for Solving Stochastic Inverse Problems
journal, January 2019

Uy, Wayne Isaac T.; Grigoriu, Mircea
SIAM Journal on Scientific Computing, Vol. 41, Issue 5
DOI: 10.1137/18m120155x

Convergence of Probability Densities using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification: Extensions to $L^p$
preprint, January 2020

Butler, Troy; Wildey, Tim; Zhang, Wenjuan
arXiv
DOI: 10.48550/arxiv.2001.04369

Learning Quantities of Interest from Dynamical Systems for Observation-Consistent Inversion
preprint, January 2020

Mattis, Steven; Steffen, Kyle Robert; Butler, Troy
arXiv
DOI: 10.48550/arxiv.2009.06918

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Similar Records

Title: Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems

Abstract

Citation Formats

A Measure-Theoretic Computational Method for Inverse Sensitivity Problems I: Method and Analysis journal, January 2011

A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion journal, January 2013

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A Computational Measure Theoretic Approach to Inverse Sensitivity Problems II: A Posteriori Error Analysis
journal, January 2012

A Measure-Theoretic Computational Method for Inverse Sensitivity Problems III: Multiple Quantities of Interest
journal, January 2014

Measure-theoretic complexity of ergodic systems
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Stochastic collocation and mixed finite elements for flow in porous media
journal, August 2008

Dimension?Adaptive Tensor?Product Quadrature
journal, August 2003

Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach
journal, September 1999

On Choosing and Bounding Probability Metrics
journal, December 2002

Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions
journal, June 2013

Statistical inverse problems: Discretization, model reduction and inverse crimes
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Uncertainty propagation using Wiener–Haar expansions
journal, June 2004

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations
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Stochastic spectral methods for efficient Bayesian solution of inverse problems
journal, June 2007

A Stochastic Collocation Algorithm with Multifidelity Models
journal, January 2014

Fusion of hard and soft information in nonparametric density estimation
journal, December 2015

On the Statistical Calibration of Physical Models: STATISTICAL CALIBRATION OF PHYSICAL MODELS
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Inverse problems: A Bayesian perspective
journal, May 2010

Variable Kernel Density Estimation
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Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs
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A multiscale preconditioner for stochastic mortar mixed finite elements
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The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
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An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions
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