Open Access
December 2016 Bayesian Solution Uncertainty Quantification for Differential Equations
Oksana A. Chkrebtii, David A. Campbell, Ben Calderhead, Mark A. Girolami
Bayesian Anal. 11(4): 1239-1267 (December 2016). DOI: 10.1214/16-BA1017


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.


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Oksana A. Chkrebtii. David A. Campbell. Ben Calderhead. Mark A. Girolami. "Bayesian Solution Uncertainty Quantification for Differential Equations." Bayesian Anal. 11 (4) 1239 - 1267, December 2016.


Published: December 2016
First available in Project Euclid: 7 September 2016

zbMATH: 1357.62108
MathSciNet: MR3577378
Digital Object Identifier: 10.1214/16-BA1017

Keywords: Bayesian numerical analysis , differential equation models , Gaussian processes , uncertainty in computer models , uncertainty quantification

Vol.11 • No. 4 • December 2016
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