Optimizing threshold-schedules for sequential approximate Bayesian computation: applications to molecular systems

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 predetermined 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 that the threshold-acceptance rate curve may be used to determine threshold schedules that avoid local optima, while balancing the need to minimise the threshold with computational efficiency. Furthermore, we provide an algorithm based upon the unscented transform, that enables the threshold-acceptance rate curve to be efficiently predicted in the case of deterministic and stochastic state space models.