  Stochastic differential equations (SDEs) provide a natural framework for
modelling intrinsic stochasticity inherent in many continuous-time physical
processes. When such processes are observed in multiple individuals or
experimental units, SDE driven mixed-effects models allow the quantification of
between (as well as within) individual variation. Performing Bayesian inference
for such models, using discrete time data that may be incomplete and subject to
measurement error is a challenging problem and is the focus of this paper. We
extend a recently proposed MCMC scheme to include the SDE driven mixed-effects
framework. Fundamental to our approach is the development of a novel construct
that allows for efficient sampling of conditioned SDEs that may exhibit
nonlinear dynamics between observation times. We apply the resulting scheme to
synthetic data generated from a simple SDE model of orange tree growth, and
real data consisting of observations on aphid numbers recorded under a variety
of different treatment regimes. In addition, we provide a systematic comparison
of our approach with an inference scheme based on a tractable approximation of
the SDE, that is, the linear noise approximation.
