Optimal projection of observations in a Bayesian setting

Handle URI:
http://hdl.handle.net/10754/627354
Title:
Optimal projection of observations in a Bayesian setting
Authors:
Giraldi, Loic ( 0000-0003-4886-5938 ) ; Le Maître, Olivier P.; Hoteit, Ibrahim ( 0000-0002-3751-4393 ) ; Knio, Omar
Abstract:
Optimal dimensionality reduction methods are proposed for the Bayesian inference of a Gaussian linear model with additive noise in presence of overabundant data. Three different optimal projections of the observations are proposed based on information theory: the projection that minimizes the Kullback–Leibler divergence between the posterior distributions of the original and the projected models, the one that minimizes the expected Kullback–Leibler divergence between the same distributions, and the one that maximizes the mutual information between the parameter of interest and the projected observations. The first two optimization problems are formulated as the determination of an optimal subspace and therefore the solution is computed using Riemannian optimization algorithms on the Grassmann manifold. Regarding the maximization of the mutual information, it is shown that there exists an optimal subspace that minimizes the entropy of the posterior distribution of the reduced model; a basis of the subspace can be computed as the solution to a generalized eigenvalue problem; an a priori error estimate on the mutual information is available for this particular solution; and that the dimensionality of the subspace to exactly conserve the mutual information between the input and the output of the models is less than the number of parameters to be inferred. Numerical applications to linear and nonlinear models are used to assess the efficiency of the proposed approaches, and to highlight their advantages compared to standard approaches based on the principal component analysis of the observations.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Physical Sciences and Engineering (PSE) Division; Earth Science and Engineering Program; Applied Mathematics and Computational Science Program
Citation:
Giraldi L, Le Maître OP, Hoteit I, Knio OM (2018) Optimal projection of observations in a Bayesian setting. Computational Statistics & Data Analysis. Available: http://dx.doi.org/10.1016/j.csda.2018.03.002.
Publisher:
Elsevier BV
Journal:
Computational Statistics & Data Analysis
KAUST Grant Number:
CRG3-2156; OSR-2016-RPP-3268
Issue Date:
18-Mar-2018
DOI:
10.1016/j.csda.2018.03.002
Type:
Article
ISSN:
0167-9473
Sponsors:
This work is supported by King Abdullah University of Science and Technology Awards CRG3-2156 and OSR-2016-RPP-3268.
Additional Links:
http://www.sciencedirect.com/science/article/pii/S0167947318300501
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Physical Sciences and Engineering (PSE) Division; Earth Science and Engineering Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorGiraldi, Loicen
dc.contributor.authorLe Maître, Olivier P.en
dc.contributor.authorHoteit, Ibrahimen
dc.contributor.authorKnio, Omaren
dc.date.accessioned2018-03-19T09:05:22Z-
dc.date.available2018-03-19T09:05:22Z-
dc.date.issued2018-03-18en
dc.identifier.citationGiraldi L, Le Maître OP, Hoteit I, Knio OM (2018) Optimal projection of observations in a Bayesian setting. Computational Statistics & Data Analysis. Available: http://dx.doi.org/10.1016/j.csda.2018.03.002.en
dc.identifier.issn0167-9473en
dc.identifier.doi10.1016/j.csda.2018.03.002en
dc.identifier.urihttp://hdl.handle.net/10754/627354-
dc.description.abstractOptimal dimensionality reduction methods are proposed for the Bayesian inference of a Gaussian linear model with additive noise in presence of overabundant data. Three different optimal projections of the observations are proposed based on information theory: the projection that minimizes the Kullback–Leibler divergence between the posterior distributions of the original and the projected models, the one that minimizes the expected Kullback–Leibler divergence between the same distributions, and the one that maximizes the mutual information between the parameter of interest and the projected observations. The first two optimization problems are formulated as the determination of an optimal subspace and therefore the solution is computed using Riemannian optimization algorithms on the Grassmann manifold. Regarding the maximization of the mutual information, it is shown that there exists an optimal subspace that minimizes the entropy of the posterior distribution of the reduced model; a basis of the subspace can be computed as the solution to a generalized eigenvalue problem; an a priori error estimate on the mutual information is available for this particular solution; and that the dimensionality of the subspace to exactly conserve the mutual information between the input and the output of the models is less than the number of parameters to be inferred. Numerical applications to linear and nonlinear models are used to assess the efficiency of the proposed approaches, and to highlight their advantages compared to standard approaches based on the principal component analysis of the observations.en
dc.description.sponsorshipThis work is supported by King Abdullah University of Science and Technology Awards CRG3-2156 and OSR-2016-RPP-3268.en
dc.publisherElsevier BVen
dc.relation.urlhttp://www.sciencedirect.com/science/article/pii/S0167947318300501en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computational Statistics & Data Analysis. 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 Computational Statistics & Data Analysis, 14 March 2018. DOI: 10.1016/j.csda.2018.03.002. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectOptimal dimensionality reductionen
dc.subjectOptimal data reductionen
dc.subjectGaussian linear modelen
dc.subjectInformation theoryen
dc.titleOptimal projection of observations in a Bayesian settingen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentEarth Science and Engineering Programen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.identifier.journalComputational Statistics & Data Analysisen
dc.eprint.versionPost-printen
dc.contributor.institutionLIMSI, CNRS, Université Paris-Saclay, Franceen
kaust.authorGiraldi, Loicen
kaust.authorHoteit, Ibrahimen
kaust.authorKnio, Omaren
kaust.grant.numberCRG3-2156en
kaust.grant.numberOSR-2016-RPP-3268en
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