Pierre Del Moral, Arnaud Doucet and Ajay Jasra just wrote a paper on ABC entitled “An Adaptive Sequential Monte Carlo Method for Approximate Bayesian Computation” that is more than welcomed as it links the ABC algorithm with their foundational SMC paper of 2006 in JRSS Series B. It thus brings a new light on the SMC ABC-PRC proposal of Sisson, Fan and Tanaka, already discussed in this post, in that both papers are based on the 2006 Series B paper idea of using a backward kernel L(z,z’) to simplify the importance weight and remove the dependence on the unknown likelihood from this weight.
The main point is that, despite a common framework, the weights of Del Moral et al. differ from those of Sisson et al. First, Del Moral et al. assume that the forward kernel K is invariant against the true target (which is a tempered version of the true posterior in sequential Monte Carlo), a choice not explicitely made in Sisson et al. (there is a mention that “Choices of K include a standard smoothing kernel (e.g., Gaussian) or a Metropolis–Hastings accept/reject step” but the target of the Metropolis–Hastings kernel is not specified—with a potential difficulty in using a proposal including a Dirac mass for importance sampling—and the kernel chosen in the toy mixture example is the “Gaussian random walk“). Second, the ABC weights of Del Moral et al. reduce to the ratio of acceptance rates for the thresholded domain, while Sisson et al. end up with the ratio of the prior densities. Only when the prior is flat do both weights coincide, but again with an invalid argument in Sisson et al. if K is not invariant against the true target.
A few comments on the paper itself:
One of the strengths of the paper is that Del Moral et al. rely on repeated simulations of the x‘s given the parameter, rather than using a single simulation given the parameter as in Sisson et al. (where this choice M=1 is considered as “the most computationally efficient“). In that perspective, each simulated parameter gets a non-zero weight that is proportional to the number of accepted x‘s. I wonder if the choice of the number M of replications could be adapted to the efficiency of the probability [of acceptation] estimation or something like that. The limiting case in M brings an exact simulation from the (tempered) target so there is a convergence principle and the stabilisation of the approximation could be assessed to control M.
The adaptivity in the ABC-SMC algorithm is dual in that (a) the Markov kernel on the parameter can be adapted in a typical PMC argument and (b) the threshold εt‘s can also be constructed on-line. The argument is to keep decreasing those thresholds slowly enough to keep a large number of accepted transitions from the previous sample. The only drawback I see is that the final value of the threshold ε is set in advance. Appart from the ultimate choice of ε = 0, which can be done in some settings as illustrated by the paper, this final value is difficult to calibrate. I also think the adaptivity criterion on the threshold ε could be Rao-Blackwellised into the sum of the importance weights rather than the indicators themselves. As noted in the paper, the ESS cannot be used per se since it is not monotonic.