## Adaptive ABC

**M**axime Lenormand, Franck Jabot and Guillaume Deffuant have just posted on arXiv a paper about a refinement of the ABC-PMC algorithm we developed with Marc Beaumont, Jean-Marie Cornuet, and Jean-Michel Marin. The authors state in their introduction that ABC-PMC

presents two shortcomings which are particularly problematic for costly to simulate complex models. First, the sequence of tolerance levels ε

_{1},…,ε_{T}has to be provided to the ABC algorithm. In practice, this implies to do preliminary simulations of the model, a step which is computationally costly for complex models. Furthermore, a badly chosen sequence of tolerance levels may inflate the number of simulations required to reach a given precision as we will see below. A second shortcoming of the PMC-ABC algorithm is that it lacks a criterion to decide whether it has converged. The final tolerance level ε_{T}may be too large for the ABC approach to satisfactorily approximate the posterior distribution of the model. Inversely, a larger ε_{T}may be sufficient to obtain a good approximation of the posterior distribution, hence sparing a number of model simulations.

shortcomings which I thought were addressed by the ABC-SMC algorithm of Pierre Del Moral, Arnaud Doucet and Ajay Jasra [not referenced in the current paper], the similar algorithm of Chris Drovandi and Tony Pettitt, and our recent paper with Jean-Michel Marin, Pierre Pudlo and Mohammed Sedki [presented at ABC in London, submitted to *Statistics and Computing* a few months ago, but alas not available on-line for unmentionable reasons linked to the growing dysfunctionality of one co-author…!]. It is correct that we did not address the choice of the ε_{t}‘s in the original paper, even though we already used an on-line selection as a quantile of the current sample of distances. In essence, given the fundamentally non-parametric nature of ABC, the tolerances ε_{t} should always be determined from the simulated samples, as regular bandwidths.

**T**he paper essentially proposes the same scheme as in Del Moral et al., before applying it to the toy example of Sisson et al. (PNAS, 2007) and to a more complex job dynamic model in central France. Were I to referee this paper, I would thus suggest that the authors incorporate a comparison with both papers of Del Moral et al. and of Drovandi and Pettitt to highlight the methodological novelty of their approach.

November 23, 2012 at 11:40 am

If you are interested, a new ABC R package called EasyABC is now available at http://easyabc.r-forge.r-project.org/ .

September 20, 2012 at 11:17 am

Many thanks for your interest in our paper. For your information, we propose a new version at http://hal.inria.fr/docs/00/72/18/85/PDF/APMC.pdf, which takes into account some of your criticisms. It includes a comparison with the methods of DelMoral et al. and Drovandi and Pettitt, and a proof that the algorithm stops.

December 11, 2011 at 12:14 am

[…] developments of Del Moral et al. and of Drovandri and Pettitt (as well as our even more recent still-un-arXived submission to Stat & Computing!). While the target is the same for all transition kernels, thus the […]

November 15, 2011 at 3:26 am

I think the stopping rule is reasonable and rather natural in the setting of their algorithm. My only concern would be if N-Na is small, then the estimate of the rare event, pacc, might be variable (and could end the algorithm prematurely). It also depends on how small the target acceptance rate, pacc_min, is.

A similar stopping rule is suggested in Drovandi and Pettitt (2011). But this is the MCMC acceptance rate (which is just the indicator under uniform prior and symmetric proposal) and is based on (N-Na)*R simulations where R gets large for small tolerances.

November 14, 2011 at 3:15 am

I think the major difference between this algorithm and other SMC ABC algorithms is that there is no repetition of the simulation step per parameter value to see if it satisfies the next tolerance (compared with the repeated simulation until the tolerance is satisfied in PMC ABC and a fixed number of simulation attempts if an MCMC kernel is used). In fact, with this algorithm, there is only (and exactly) N-Na simulations per target. This would be at the expense of requiring many tolerances (target distributions) since (new) particles above the next tolerance are still kept. If all N-Na simulations are above the next tolerance then the algorithm gives up as it has not moved anywhere.

It would seem that N-Na must not be too small otherwise a poor estimate of pacc may be obtained (especially for small tolerances, i.e. rare event) and the algorithm could end prematurely.

I think the authors should focus on this difference to see if this trade-off gives any improvement.

November 14, 2011 at 7:54 am

Thank you, Chris, for the comments. I missed that distinctive point. What do you make of the stopping rule?