## ABC for parameter inference and model selection in dynamical systems

**A**nother 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).

**T**he 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 * K_{t}*‘s remaining the same across t, in contrast with our PMC motivation for tuning

*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*

**K**_{t}*, which should not be the case, since this is a simulation parameter that is unrelated with the statistical problem.*

**K**_{t}**A**nother 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 * K_{t }*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

*corresponds to using the flat prior (see Figure 1) and ultimately to the ABC rejection algorithm.*

**K**_{t }

May 6, 2011 at 7:17 pm

[…] than 5 million years as thought previously). Tina Toni talked about the application of ABC-SMC and ABC model choice to complex biochemical dynamics. Pierre Pudlo and Mohammed Sedki introduced the new ABC-SMC scheme […]

January 31, 2011 at 12:17 am

[…] there are cases where summary statistics are not required yet ABC is still used (see eg. Toni et al). Conversely, likelihood based inference is sometimes done based on summary statistics. For […]

January 8, 2009 at 11:18 am

Thanks for the comments Christian and apologies again for sending an incorrect draft. I’d just like to point out that Mark Beaumont ended up refereeing the Toni et al paper which originally used the Sisson et al (2007) PRC ABC algorithm. Mark pointed out that this algorithm was incorrect and suggested a fix which we modified to get this final version. It’s therefore not surprising that there are parallels with this paper and Beaumont et al (2008).

David

January 8, 2009 at 8:23 am

[…] Xi’an’s Og An attempt at bloggin, from scratch… « ABC for parameter inference and model selection in dynamical systems […]