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Bayes linear analysis for ordinary differential equations

Jones, Matthew; Goldstein, Michael; Randell, David; Jonathan, Philip

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Matthew Jones

David Randell

Philip Jonathan


Differential equation models are used in a wide variety of scientific fields to describe the behaviour of physical systems. Commonly, solutions to given systems of differential equations are not available in closed-form; in such situations, the solution to the system is generally approximated numerically. The numerical solution obtained will be systematically different from the (unknown) true solution implicitly defined by the differential equations. Even if it were known, this true solution would be an imperfect representation of the behaviour of the real physical system that it was designed to represent. A Bayesian framework is proposed which handles all sources of numerical and structural uncertainty encountered when using ordinary differential equation (ODE) models to represent real-world processes. The model is represented graphically, and the graph proves to be useful tool, both for deriving a full prior belief specification and for inferring model components given observations of the real system. A general strategy for modelling the numerical discrepancy induced through choice of a particular solver is outlined, in which the variability of the numerical discrepancy is fixed to be proportional to the length of the solver time-step and a grid-refinement strategy is used to study its structure in detail. A Bayes linear adjustment procedure is presented, which uses a junction tree derived from the originally specified directed graphical model to propagate information efficiently between model components, lessening the computational demands associated with the inference. The proposed framework is illustrated through application to two examples: a model for the trajectory of an airborne projectile moving subject to gravity and air resistance, and a model for the coupled motion of a set of ringing bells and the tower which houses them.


Jones, M., Goldstein, M., Randell, D., & Jonathan, P. (2021). Bayes linear analysis for ordinary differential equations. Computational Statistics & Data Analysis, 161, Article 107228.

Journal Article Type Article
Acceptance Date Mar 13, 2021
Online Publication Date Mar 24, 2021
Publication Date 2021-09
Deposit Date Mar 31, 2022
Publicly Available Date Mar 31, 2022
Journal Computational Statistics & Data Analysis
Print ISSN 0167-9473
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 161
Article Number 107228


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