  Sequential Monte Carlo squared (SMC$^2$) methods can be used for parameter
inference of intractable likelihood state-space models. These methods replace
the likelihood with an unbiased particle filter estimator, similarly to
particle Markov chain Monte Carlo (MCMC). As with particle MCMC, the efficiency
of SMC$^2$ greatly depends on the variance of the likelihood estimator, and
therefore on the number of state particles used within the particle filter. We
introduce novel methods to adaptively select the number of state particles
within SMC$^2$ using the expected squared jumping distance to trigger the
adaptation, and modifying the exchange importance sampling method of
\citet{Chopin2012a} to replace the current set of state particles with the new
set of state particles. The resulting algorithm is fully automatic, and can
significantly improve current methods. Code for our methods is available at
https://github.com/imkebotha/adaptive-exact-approximate-smc.
