  The challenging problem of conducting fully Bayesian inference for the
reaction rate constants governing stochastic kinetic models (SKMs) is
considered. Given the challenges underlying this problem, the Markov jump
process representation is routinely replaced by an approximation based on a
suitable time discretisation of the system of interest. 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.
