  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 and show the parameters in these models can be realistically
estimated from simulated data. Preliminary analyses based on the Fisher
Information Matrix of the model can guide the implementation of Bayesian
inference. We show that this parameter sensitivity analysis can predict which
parameters are practically identifiable. Several Markov chain Monte Carlo
algorithms are compared, with our results suggesting a parallel tempering
algorithm consistently gives the best approach for these systems, which are
shown to frequently exhibit multi-modal posterior distributions.
