  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.
