On optimal kernel in ABC SMC

Sequential Monte Carlo (SMC) approaches have become work horses in approximate Bayesian computation (ABC). Here we discuss how to construct the perturbation kernels that are required in ABC-SMC approaches, in order to construct a set of distributions that start out from a suitably defined prior and converge towards the unknown posterior. We derive optimality criteria for different kernels, which are based on the Kullback-Leibler divergence between intermediate distributions and the posterior distribution. We will show that for many complicated posterior distributions locally adapted kernels tend to show the best performance. In cases where it is possible to estimate the Fisher information we can construct particularly efficient perturbation kernels. We find that the added moderate cost of adapting kernel functions is easily regained in terms of the higher acceptance rate. We demonstrate the computational efficiency gains in a range of toy-examples which illustrate some of the challenges faced in real-world applications of ABC, before turning to a demanding parameter inference problem for a dynamical system, which highlights the large gains in efficiency that can be gained from choice of optimal models. We conclude with a general discussion of rational choice of perturbation kernels in ABC-SMC settings.