Bayesian Inference for Diffusion-Driven Mixed-Effects Models

Abstract

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 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 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 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.

Bayesian Inference for Stochastic Differential Equation Mixed Effects Models of a Tumour Xenography Study

“…For instance, Donnet et al (2010) found that a stochastic differential equation (SDE) version of the Gompertz growth model is superior to its non-linear deterministic mixed model counterpart, for prediction of the body weight of growing chickens. Donnet and Samson (2013) reported similar findings from pharmacokinetic experiments and Whitaker et al (2017) provide more recent references to inference for SDEMEMs. What prevents more widespread application of SDEMEMs is that inference for non-linear SDE models is overall challenging, even when not considering random effects and measurement errors, because SDEs generally have intractable likelihood functions (Fuchs, 2013).…”

“…A recent review of Bayesian inference methods for SDEMEMs can be found in Whitaker et al . (). It is important to note that Markov chain Monte Carlo algorithms can be constructed to sample from the exact posterior of θ , for models admitting a non‐negative unbiased estimator of p ( y | θ ) (Beaumont, ; Andrieu and Roberts, ).…”

“…Donnet and Samson () reported similar findings from pharmacokinetic experiments and Whitaker et al . () provide more recent references to inference for SDEMEMs. What prevents more widespread application of SDEMEMs is that inference for non‐linear SDE models is overall challenging, even when not considering random effects and measurement errors, because SDEs generally have intractable likelihood functions (Fuchs, ).…”

Comprehensive Review of Models and Methods for Inferences in Bio-Chemical Reaction Networks

“…Nested Bayesian sampling is used in (Pullen and Morris, 2014) to compute marginal likelihoods to compare or rank several competing models. MCMC sampling for mixed-effects SDE models is considered in (Whitaker et al, 2017). In order to overcome ill-conditioned least squares (LS) data fitting and numerical instability, bootstrapped MC procedure based on diffusion and LNA was studied in (Lindera and Rempala, 2015).…”

“…Models with memory described by delay differential equations (DDEs) are investigated in (Zhan et al, 2014). Mixed-effect models assume multiple instances of SDE based models to evaluate statistical variations between and within these models (Whitaker et al, 2017).…”