Bayesian inference for stochastic differential mixed effects models
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 mixedeffects models allow the quantification of both between and 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 thesis. Since, in general, no closed form expression exists for the transition densities of the SDE of interest, a widely adopted solution works with the Euler Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. These approximations can be made arbitrarily accurate by introducing intermediate time points between observations. Integrating over the uncertainty associated with the process at these time points necessitates the use of computationally intensive algorithms such as Markov chain Monte Carlo (MCMC) We extend a recently proposed MCMC scheme to include the SDE driven mixed effects framework. Key to the development of an efficient inference scheme is the ability to generate discrete time realisations of the latent process between observation times. Such realisations are typically termed diffusion bridges. By partitioning the SDE into two parts, one that accounts for nonlinear dynamics in a deterministic way, and another as a residual stochastic process, we develop a class of novel constructs that bridge the residual process via a linear approximation. In addition, we adapt a recently proposed construct to a partial and noisy observation regime. We compare the performance of each new construct with a number of existing approaches, using three applications: a simple birth death process, a Lotka Volterra model and a model for aphid growth. We incorporate the best performing bridge construct within an MCMC scheme to determine the posterior distribution of the model parameters. This methodology is then applied 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. Finally, 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.