  Recently-proposed particle MCMC methods provide a flexible way of performing
Bayesian inference for parameters governing stochastic kinetic models defined
as Markov (jump) processes (MJPs). Each iteration of the scheme requires an
estimate of the marginal likelihood calculated from the output of a sequential
Monte Carlo scheme (also known as a particle filter). Consequently, the method
can be extremely computationally intensive. We therefore aim to avoid most
instances of the expensive likelihood calculation through use of a fast
approximation. We consider two approximations: the chemical Langevin equation
diffusion approximation (CLE) and the linear noise approximation (LNA). Either
an estimate of the marginal likelihood under the CLE, or the tractable marginal
likelihood under the LNA can be used to calculate a first step acceptance
probability. Only if a proposal is accepted under the approximation do we then
run a sequential Monte Carlo scheme to compute an estimate of the marginal
likelihood under the true MJP and construct a second stage acceptance
probability that permits exact (simulation based) inference for the MJP. We
therefore avoid expensive calculations for proposals that are likely to be
rejected. We illustrate the method by considering inference for parameters
governing a Lotka-Volterra system, a model of gene expression and a simple
epidemic process.
