A Laplace method for under-determined Bayesian optimal experimental designs

Handle URI:
http://hdl.handle.net/10754/557227
Title:
A Laplace method for under-determined Bayesian optimal experimental designs
Authors:
Long, Quan ( 0000-0002-0329-9437 ) ; Scavino, Marco ( 0000-0001-5114-853X ) ; Tempone, Raul ( 0000-0003-1967-4446 ) ; Wang, Suojin
Abstract:
In Long et al. (2013), a new method based on the Laplace approximation was developed to accelerate the estimation of the post-experimental expected information gains (Kullback–Leibler divergence) in model parameters and predictive quantities of interest in the Bayesian framework. A closed-form asymptotic approximation of the inner integral and the order of the corresponding dominant error term were obtained in the cases where the parameters are determined by the experiment. In this work, we extend that method to the general case where the model parameters cannot be determined completely by the data from the proposed experiments. We carry out the Laplace approximations in the directions orthogonal to the null space of the Jacobian matrix of the data model with respect to the parameters, so that the information gain can be reduced to an integration against the marginal density of the transformed parameters that are not determined by the experiments. Furthermore, the expected information gain can be approximated by an integration over the prior, where the integrand is a function of the posterior covariance matrix projected over the aforementioned orthogonal directions. To deal with the issue of dimensionality in a complex problem, we use either Monte Carlo sampling or sparse quadratures for the integration over the prior probability density function, depending on the regularity of the integrand function. We demonstrate the accuracy, efficiency and robustness of the proposed method via several nonlinear under-determined test cases. They include the designs of the scalar parameter in a one dimensional cubic polynomial function with two unidentifiable parameters forming a linear manifold, and the boundary source locations for impedance tomography in a square domain, where the unknown parameter is the conductivity, which is represented as a random field.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
A Laplace method for under-determined Bayesian optimal experimental designs 2015, 285:849 Computer Methods in Applied Mechanics and Engineering
Journal:
Computer Methods in Applied Mechanics and Engineering
Issue Date:
17-Dec-2014
DOI:
10.1016/j.cma.2014.12.008
Type:
Article
ISSN:
00457825
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0045782514004873
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorLong, Quanen
dc.contributor.authorScavino, Marcoen
dc.contributor.authorTempone, Raulen
dc.contributor.authorWang, Suojinen
dc.date.accessioned2015-06-18T07:24:54Zen
dc.date.available2015-06-18T07:24:54Zen
dc.date.issued2014-12-17en
dc.identifier.citationA Laplace method for under-determined Bayesian optimal experimental designs 2015, 285:849 Computer Methods in Applied Mechanics and Engineeringen
dc.identifier.issn00457825en
dc.identifier.doi10.1016/j.cma.2014.12.008en
dc.identifier.urihttp://hdl.handle.net/10754/557227en
dc.description.abstractIn Long et al. (2013), a new method based on the Laplace approximation was developed to accelerate the estimation of the post-experimental expected information gains (Kullback–Leibler divergence) in model parameters and predictive quantities of interest in the Bayesian framework. A closed-form asymptotic approximation of the inner integral and the order of the corresponding dominant error term were obtained in the cases where the parameters are determined by the experiment. In this work, we extend that method to the general case where the model parameters cannot be determined completely by the data from the proposed experiments. We carry out the Laplace approximations in the directions orthogonal to the null space of the Jacobian matrix of the data model with respect to the parameters, so that the information gain can be reduced to an integration against the marginal density of the transformed parameters that are not determined by the experiments. Furthermore, the expected information gain can be approximated by an integration over the prior, where the integrand is a function of the posterior covariance matrix projected over the aforementioned orthogonal directions. To deal with the issue of dimensionality in a complex problem, we use either Monte Carlo sampling or sparse quadratures for the integration over the prior probability density function, depending on the regularity of the integrand function. We demonstrate the accuracy, efficiency and robustness of the proposed method via several nonlinear under-determined test cases. They include the designs of the scalar parameter in a one dimensional cubic polynomial function with two unidentifiable parameters forming a linear manifold, and the boundary source locations for impedance tomography in a square domain, where the unknown parameter is the conductivity, which is represented as a random field.en
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0045782514004873en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 17 December 2014. DOI: 10.1016/j.cma.2014.12.008en
dc.subjectBayesian statisticsen
dc.subjectOptimal experimental designen
dc.subjectInformation gainen
dc.subjectLaplace approximationen
dc.subjectMonte Carlo samplingen
dc.subjectSparse quadratureen
dc.titleA Laplace method for under-determined Bayesian optimal experimental designsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalComputer Methods in Applied Mechanics and Engineeringen
dc.eprint.versionPost-printen
dc.contributor.institutionICES, University of Texas at Austin, Austin, 78712-1299, USAen
dc.contributor.institutionInstituto de Estadística (IESTA), Universidad de la República, Montevideo, Uruguayen
dc.contributor.institutionDepartment of Statistics, Texas A & M University, College Station, TX, 77843, USAen
kaust.authorLong, Quanen
kaust.authorTempone, Raulen
kaust.authorScavino, Marcoen
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