Adaptivity for ABC algorithms: the ABC-PMC scheme

Sequential techniques can be added to the approximate Bayesian computation (ABC) algorithm to enhance its efficiency. Sisson et al. (2007) introduced the ABC-PRC algorithm to improve upon existing ABC-MCMC algorithms. While the ABC-PRC method is based upon the theoretical developments of Del Moral et al. (2006), the application to the ABC setting induces a bias in the approximation to the posterior distribution of interest. It is however possible to devise an alternative version based on genuine importance sampling arguments that we call ABC-PMC in connection with the population Monte Carlo method introduced in Cappe et al. (2004). This algorithm is simpler than the ABC-PRC algorithm, it does not suffer from the original bias, and it includes an automatic scaling of the forward kernel. Moreover, when applied to a population genetics example, its efficiency compares favourably with two standard ABC algorithms.