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Research Papers

A Linear Regression Framework for the Verification of Bayesian Model Calibration Algorithms

[+] Author and Article Information
Jerry A. McMahan, Jr.

Digimarc Corporation,
Beaverton, OR 97008
e-mail: jerry.mcmahan@digimarc.com

Brian J. Williams

Statistical Sciences Group (CCS-6),
Los Alamos National Laboratory,
Los Alamos, NM 87545
e-mail: brianw@lanl.gov

Ralph C. Smith

Mem. ASME
Department of Mathematics,
North Carolina State University,
Raleigh, NC 27695
e-mail: rsmith@ncsu.edu

Nicholas Malaya

Mem. ASME
Department of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712
e-mail: nick@ices.utexas.edu

Manuscript received November 20, 2016; final manuscript received August 14, 2017; published online September 12, 2017. Editor: Ashley F. Emery. The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Verif. Valid. Uncert 2(2), 021006 (Sep 12, 2017) (14 pages) Paper No: VVUQ-16-1031; doi: 10.1115/1.4037705 History: Received November 20, 2016; Revised August 14, 2017

We describe a framework for the verification of Bayesian model calibration routines. The framework is based on linear regression and can be configured to verify calibration to data with a range of observation error characteristics. The framework is designed for efficient implementation and is suitable for verifying code intended for large-scale problems. We propose an approach for using the framework to verify Markov chain Monte Carlo (MCMC) software by combining it with a nonparametric test for distribution equality based on the energy statistic. Our matlab-based reference implementation of the framework is shown to correctly distinguish between output obtained from correctly and incorrectly implemented MCMC routines. Since correctness of output from an MCMC software depends on choosing settings appropriate for the problem-of-interest, our framework can potentially be used for verifying such settings.

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Figures

Grahic Jump Location
Fig. 1

Comparison of DRAM-computed densities with the analytically computed densities for case 3 (β̂,λ̂,ϕ̂ calibrated) with AR(1) correlated observation error and noninformative prior: (a) β̂1 correct code, (b) β̂1 incorrect code, (c) β̂2 correct code, and (d) β̂2 incorrect code

Grahic Jump Location
Fig. 2

Comparison of DRAM-computed densities with the analytically computed densities for case 3 (β̂,λ̂,ϕ̂ calibrated) with AR(1) correlated observation error and noninformative prior: (a) λ̂ correct code, (b) λ̂ incorrect code, (c) ϕ̂ correct code, and (d) ϕ̂ incorrect code

Grahic Jump Location
Fig. 3

Comparison of DRAM-computed densities with the analytically computed densities for case 3 (β̂,λ̂,ϕ̂ calibrated) with AR(1) correlated observation error and Gaussian prior: (a) β̂1 correct code, (b) β̂1 incorrect code, (c) β̂2 correct code, and (d) β̂2 incorrect code

Grahic Jump Location
Fig. 4

Comparison of DRAM-computed densities with the analytically computed densities for case 3 (β̂,λ̂,ϕ̂ calibrated) with AR(1) correlated observation error and Gaussian prior: (a) λ̂ correct code, (b) λ̂ incorrect code, (c) ϕ̂ correct code, and (d) ϕ̂ incorrect code

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