  Fitting stochastic kinetic models represented by Markov jump processes within
the Bayesian paradigm is complicated by the intractability of the observed data
likelihood. There has therefore been considerable attention given to the design
of pseudo-marginal Markov chain Monte Carlo algorithms for such models.
However, these methods are typically computationally intensive, often require
careful tuning and must be restarted from scratch upon receipt of new
observations. Sequential Monte Carlo (SMC) methods on the other hand aim to
efficiently reuse posterior samples at each time point. Despite their appeal,
applying SMC schemes in scenarios with both dynamic states and static
parameters is made difficult by the problem of particle degeneracy. A
principled approach for overcoming this problem is to move each parameter
particle through a Metropolis-Hastings kernel that leaves the target invariant.
This rejuvenation step is key to a recently proposed SMC$^2$ algorithm, which
can be seen as the pseudo-marginal analogue of an idealised scheme known as
iterated batch importance sampling. Computing the parameter weights in SMC$^2$
requires running a particle filter over dynamic states to unbiasedly estimate
the intractable observed data likelihood contributions at each time point. In
this paper, we propose to use an auxiliary particle filter inside the SMC$^2$
scheme. Our method uses two recently proposed constructs for sampling
conditioned jump processes and we find that the resulting inference schemes
typically require fewer state particles than when using a simple bootstrap
filter. Using two applications, we compare the performance of the proposed
approach with various competing methods, including two global MCMC schemes.
