  Approximate Bayesian computation (ABC) is a well-established family of Monte
Carlo methods for performing approximate Bayesian inference in the case where
an ``implicit'' model is used for the data: when the data model can be
simulated, but the likelihood cannot easily be pointwise evaluated. A
fundamental property of standard ABC approaches is that the number of Monte
Carlo points required to achieve a given accuracy scales exponentially with the
dimension of the data. Prangle et al. (2018) proposes a Markov chain Monte
Carlo (MCMC) method that uses a rare event sequential Monte Carlo (SMC)
approach to estimating the ABC likelihood that avoids this exponential scaling,
and thus allows ABC to be used on higher dimensional data. This paper builds on
the work of Prangle et al. (2018) by using the rare event SMC approach within
an SMC algorithm, instead of within an MCMC algorithm. The new method has a
similar structure to SMC$^{2}$ (Chopin et al., 2013), and requires less tuning
than the MCMC approach. We demonstrate the new approach, compared to existing
ABC-SMC methods, on a toy example and on a duplication-divergence random graph
model used for modelling protein interaction networks.
