Correlated pseudo-marginal schemes for time-discretised stochastic kinetic models
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Performing fully Bayesian inference for the reaction rate constants governing stochastic kinetic models (SKMs) is a challenging problem. Consequently, the 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 the number of reaction events as Gaussian leads to the chemical Langevin equation (also known as the diffusion approximation). 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. 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 likelihood estimate at the current and proposed values of the unknown parameters. We extend this approach to the time discretised SKM framework by correlating the innovations that drive the auxiliary particle filter. We find that the resulting approach offers substantial gains in efficiency over a standard implementation.
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