  The likelihood-free sequential Approximate Bayesian Computation (ABC)
algorithms, are increasingly popular inference tools for complex biological
models. Such algorithms proceed by constructing a succession of probability
distributions over the parameter space conditional upon the simulated data
lying in an $\epsilon$--ball around the observed data, for decreasing values of
the threshold $\epsilon$. While in theory, the distributions (starting from a
suitably defined prior) will converge towards the unknown posterior as
$\epsilon$ tends to zero, the exact sequence of thresholds can impact upon the
computational efficiency and success of a particular application. In
particular, we show here that the current preferred method of choosing
thresholds as a pre-determined quantile of the distances between simulated and
observed data from the previous population, can lead to the inferred posterior
distribution being very different to the true posterior. Threshold selection
thus remains an important challenge. Here we propose an automated and adaptive
method that allows us to balance the need to minimise the threshold with
computational efficiency. Moreover, our method which centres around predicting
the threshold - acceptance rate curve using the unscented transform, enables us
to avoid local minima - a problem that has plagued previous threshold schemes.
