Accelerating pseudo marginal Metropolis-Hastings schemes for stochastic kinetic models

Performing fully Bayesian inference for the rate constants governing stochastic kinetic models is a challenging problem. The most natural Markov jump process representation is routinely replaced by an approximation based on a suitable time discretisation of the system of interest. For example, assuming that instantaneous rates of reactions are constant over a small time period leads to a poisson approximation of the number of reaction events in the interval. Further approximating reaction events as Gaussian leads to the chemical Langevin equation. Improving the accuracy of these schemes amounts to using an ever finer discretisation level, which in the context of the inference problem, requires integrating over the uncertainty in the process at a predetermined number of intermediate times between observations. Pseudo-marginal Metropolis-Hastings schemes are increasingly used, since for a given discretisation level, the observed data likelihood can be unbiasedly estimated using a particle filter. When observations are particularly informative, an auxiliary particle filter can be implemented, by employing an appropriate construct to push the state particles towards the observations in a sensible way. The aim of this project is as follows: Recent work in state-space settings has shown how the pseudo-marginal approach can be made much more efficient by correlating the underlying pseudo random numbers used to form the estimate of likelihood at the current and proposed values of the unknown parameters. This idea will be explored in the context of time discretised stochastic kinetic models and other stochastic differential equations, in particular the chemical Langevin equation and poisson leap methods. This will require getting to grips with these approximations to the Markov jump process representation of a reaction network. We will seek to further increase computational efficiency through the use of delayed acceptance (DA) methods, and in particular examine the use of correlation between different stages in the DA algorithm. We will compare and contrast these methods with existing approaches for biologically interesting applications e.g. models of epidemics, predator-prey interaction and intra-cellular processes.