  Bayesian experimental design (BED) is a framework that uses statistical
models and decision making under uncertainty to optimise the cost and
performance of a scientific experiment. Sequential BED, as opposed to static
BED, considers the scenario where we can sequentially update our beliefs about
the model parameters through data gathered in the experiment. A class of models
of particular interest for the natural and medical sciences are implicit
models, where the data generating distribution is intractable, but sampling
from it is possible. Even though there has been a lot of work on static BED for
implicit models in the past few years, the notoriously difficult problem of
sequential BED for implicit models has barely been touched upon. We address
this gap in the literature by devising a novel sequential design framework for
parameter estimation that uses the Mutual Information (MI) between model
parameters and simulated data as a utility function to find optimal
experimental designs, which has not been done before for implicit models. Our
approach uses likelihood-free inference by ratio estimation to simultaneously
estimate posterior distributions and the MI. During the sequential BED
procedure we utilise Bayesian optimisation to help us optimise the MI utility.
We find that our framework is efficient for the various implicit models tested,
yielding accurate parameter estimates after only a few iterations.
