## Bayesian Analysis

- Bayesian Anal.
- Volume 11, Number 4 (2016), 1239-1267.

### Bayesian Solution Uncertainty Quantification for Differential Equations

Oksana A. Chkrebtii, David A. Campbell, Ben Calderhead, and Mark A. Girolami

#### 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.

#### Article information

**Source**

Bayesian Anal. Volume 11, Number 4 (2016), 1239-1267.

**Dates**

First available in Project Euclid: 7 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ba/1473276259

**Digital Object Identifier**

doi:10.1214/16-BA1017

**Zentralblatt MATH identifier**

1357.62108

**Keywords**

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

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Chkrebtii, Oksana A.; Campbell, David A.; Calderhead, Ben; Girolami, Mark A. Bayesian Solution Uncertainty Quantification for Differential Equations. Bayesian Anal. 11 (2016), no. 4, 1239--1267. doi:10.1214/16-BA1017. https://projecteuclid.org/euclid.ba/1473276259

#### See also

- Related item: Martin Lysy (2016). Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Anal. Vol. 11, Iss. 4, 1269–1273. Digital Object Identifier: doi:10.1214/16-BA1036
- Related item: Sarat C. Dass (2016). Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Anal. Vol. 11, Iss. 4, 1275–1277. Digital Object Identifier: doi:10.1214/16-BA1037
- Related item: Bani K. Mallick, Keren Yang, Nilabja Guha, Yalchin Efendiev (2016). Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Anal. Vol. 11, Iss. 4, 1279–1284. Digital Object Identifier: doi:10.1214/16-BA1038
- Related item: François-Xavier Briol, Jon Cockayne, Onur Teymur, William Weimin Yoo, Jon Cockayne, Michael Schober, Philipp Hennig (2016). Contributed Discussion on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Anal. Vol. 11, Iss. 4, 1285–1293. Digital Object Identifier: doi:10.1214/16-BA1017A
- Related item: Oksana A. Chkrebtii, David A. Campbell, Ben Calderhead, Mark A. Girolami (2016). Rejoinder. Bayesian Anal. Vol. 11, Iss. 4, 1295–1299. Digital Object Identifier: doi:10.1214/16-BA1017REJ

#### Supplemental materials

- Supplementary Material for “Bayesian Solution Uncertainty Quantification for Differential Equations”. Digital Object Identifier: doi:10.1214/16-BA1017SUPP