ABC for parameter inference and model selection in dynamical systems
Another paper on ABC that is about to appear [in Journal of the Royal Society Interface, 2009] is “Approximate Bayesian Computation scheme for parameter inference and model selection in dynamical systems” by Toni, Welch, Strelkowa, Ipsen and Stumpf. (It is not available on-line, as far as I can see.) This paper is quite interesting in particular in that its central ABC algorithm parallels (with differences, of course!) our ABC-PMC paper by Beaumont et al. (2008), which was completed and submitted a bit later (and obviously independently). Just like ABC-PMC, the ABC SMC algorithm developped by Toni et al. (2009) is using a sequence of populations (or samples), (Markov) transition kernels, and importance weights (rather than SMC justifications) where the unavailable likelihood is (approximately) estimated by the indicator of the tolerance zone or an average of indicators as in Marjoram et al. (2003).
The paper is intended for a broad audience in that it also surveys both ABC methods and Bayesian model choice principles, incorporating the simulation of model indicators within the ABC framework as in our paper by Grelaud et al. (2008), where we should have quoted Toni et al. (2009), had we been aware of the contents of the paper… (This paper on ABC for model choice in Ising models has been tentatively accepted by Bayesian Analysis, and we will definitely acknowledge the paper and its model selection algorithm in the revision.) The main bulk of the paper is dedicated to the analysis of ODEs with two and four parameters, using uniforms as transition kernels. The adaptivity (or sequentiality) of the ABC SMC algorithm is restricted to a progressive reduction of the tolerance, the kernels Kt‘s remaining the same across t, in contrast with our PMC motivation for tuning Kt to the target. The Bayes factors are observed to be sensitive to the choice of the prior distributions (fair), of the tolerance levels (also fair), and to the variances of the kernel Kt, which should not be the case, since this is a simulation parameter that is unrelated with the statistical problem.
Another interesting part of the paper is its appendix, made of a justification for the ABC SMC algorithm and of a comparison with Sisson et al.’s (2007) ABC PRC. As shown by my attached preliminary notes below, I had difficulties with the finer details of the ABC SMC weights as presented in step S2.2 of the algorithm, which is presumably a typo due to a lack of precision in the notations, given that the algorithmic developments in the appendix are correct.
The comparison with Sisson et al.’s (2007) is operated on the normal mixture toy example and it shows a bias in the variance of the ABC PRC output when using for the forward kernel Kt a uniform random walk with a small range, while a larger range recovers the right posterior variance. This is in line with our experiment in that using a large scale in Kt corresponds to using the flat prior (see Figure 1) and ultimately to the ABC rejection algorithm.