Bayesian inference for stochastic oscillatory systems using the phase-corrected linear noise approximation

Likelihood-based inference in stochastic non-linear dynamical systems, such as those found in chemical reaction networks and biological clock systems, is inherently complex and has largely been limited to small and unrealistically simple systems. Recent advances in analytically tractable approximations to the underlying conditional probability distributions enable long-term dynamics to be accurately modelled, and make the large number of model evaluations required for exact Bayesian inference much more feasible. We propose a new methodology for inference in stochastic non-linear dynamical systems exhibiting oscillatory behaviour that can be applied even if the system involves a large number of variables and unknown parameters. and show the parameters in these models can be realistically estimated from simulated data. We use preliminary analyses based on the Fisher Information Matrix of the model to guide the implementation of Bayesian inference. We show that this parameter sensitivity analysis can predict which parameters are practically identifiable. A parallel tempering algorithm is used to provide the flexibility required to explore the posterior distribution of the model parameters, which often exhibit multi-modal posterior distributions.