threshold schedules for ABC-SMC (reply from the authors)
In our experience the problems that we seek to address arise routinely if adaptive quantile schedules are employed and are not due to poor initial exploration of the parameter space. Rather, when sub-optimal quantiles are chosen we observe what has often been called the “survival of the fattest”. Here a finite population of particles is accepted into regions that satisfy some less stringent cost function (e.g. simulated annealing at high temperature) easily but thereby drifts (in the genetic sense) away from better solutions that would also satisfy tighter thresholds. We do not claim to be the first to point this out; but we believe that depending on the problem any quantile method will be susceptible to this. A similar point has been made, perhaps more beautifully, by Calvet and Czellar (2012).
The toy example is maybe slightly misleading in the sense that parameter-space and data-space can be mapped bijectively. It is not, however, fanciful: it is an illustration of “survival of the fattest”. And it also demonstrates that reliance on a fixed quantile (say 10%) for any type of problem is likely to result in systematically biased, plainly incorrect “posteriors” in some if not most cases.
We do not claim that the optimal ε is zero. But clearly it is not the one dictated by some fixed computational cost you want to get away with (as is the case when people take the best n out of N simulations). What we try to argue is that the threshold should be set based on the problem under consideration. Therefore the choice of ε should make use of the hypothetical acceptance curve, ℵ.
Clearly we do not know ℵ but show that the UT is one way of second-guessing its shape. That each guess depends on the present population is both obvious and necessary.
A CDF-based estimate for ℵ using the empirical distances from the previous population would ignore the role of the perturbation kernel that we employ (and which should also be problem-specific). This we found, and by hindsight should have been obvious, has huge impact on the efficiency, in particular the validity of ℵ. Of course, if we already had the CDF of the present target population (ie the whole set of perturbed particles) we would be fine. But knowing the answer makes any question look easy. But getting an answer to the types of problems that interest us is definitely obtained more quickly when we use the UT.
Maybe the problems that we are dealing with in our “day-jobs” (in the analysis of biomolecular and non-linear dynamical systems) make the shortcomings of the “quantile-method” more readily apparent; against this background the proposed approach is perhaps more easily understood.