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Bayesian Solution Uncertainty Quantification for Differential Equations

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Title: Bayesian Solution Uncertainty Quantification for Differential Equations
Authors: Chkrebtii, OA
Campbell, DA
Calderhead, B
Girolami, MA
Item Type: Journal Article
Abstract: We explore probability modelling of discretization uncertainty for system states defined implicitly by ordinary or partial differential equations. Accounting for this uncertainty can avoid posterior under-coverage when likelihoods are constructed from a coarsely discretized approximation to system equations. A formalism is proposed for inferring a fixed but a priori unknown model trajectory through Bayesian updating of a prior process conditional on model information. A one-step-ahead sampling scheme for interrogating the model is described, its consistency and first order convergence properties are proved, and its computational complexity is shown to be proportional to that of numerical explicit one-step solvers. Examples illustrate the flexibility of this framework to deal with a wide variety of complex and large-scale systems. Within the calibration problem, discretization uncertainty defines a layer in the Bayesian hierarchy, and a Markov chain Monte Carlo algorithm that targets this posterior distribution is presented. This formalism is used for inference on the JAK-STAT delay differential equation model of protein dynamics from indirectly observed measurements. The discussion outlines implications for the new field of probabilistic numerics.
Issue Date: 7-Sep-2016
Date of Acceptance: 1-Sep-2016
URI: http://hdl.handle.net/10044/1/49870
DOI: https://dx.doi.org/10.1214/16-BA1017
ISSN: 1936-0975
Publisher: International Society for Bayesian Analysis
Start Page: 1239
End Page: 1267
Journal / Book Title: Bayesian Analysis
Volume: 11
Issue: 4
Copyright Statement: © 2016 International Society for Bayesian Analysis.
Keywords: Science & Technology
Physical Sciences
Mathematics, Interdisciplinary Applications
Statistics & Probability
Mathematics
Bayesian numerical analysis
uncertainty quantification
Gaussian processes
differential equation models
uncertainty in computer models
CHAIN MONTE-CARLO
PARAMETER-ESTIMATION
INVERSE PROBLEMS
PROFILE LIKELIHOOD
MODEL-REDUCTION
SYSTEMS
IDENTIFICATION
TOMOGRAPHY
ERROR
ODES
stat.ME
0104 Statistics
Publication Status: Published
Appears in Collections:Mathematics
Statistics



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