Adaptivity for ABC algorithms: the ABC-PMC scheme

Sequential techniques can be adapted to the ABC algorithm to enhance its efficiency. For instance, when Sisson et al. (2007) introduced the ABC-PRC algorithm, the goal was to improve upon existing ABC-MCMC algorithms (Marjoram et al., 2003). While the ABC-PRC method is based upon the theoretical developments of Del Moral et al. (2006), the application to the setting of approximate Bayesian computation induces a bias in the approximation to the posterior distribution of interest, as we demonstrate in this paper via both theoretical reasoning and experimental results. 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 Cappé et al. (2004). Furthermore, the ABC-PMC algorithm is simpler than the ABC-PRC algorithm in that it does not require any backward transition kernel and proposes an automatic scaling of the forward kernel. In this paper, we demonstrate the applicability of ABC-PMC and compare its performances with ABC-PRC.