## Another ABC paper

“One aim is to extend the approach of Sisson et al. (2007) to provide an algorithm that is robust to implement.”

C.C. Drovandi & A.N. Pettitt

A paper by Drovandi and Pettit appeared in the Early View section of Biometrics. It uses a combination of particles and of MCMC moves to adapt to the true target, with an acceptance probability

$\min\left\{1,\dfrac{\pi(\theta^*)q(\theta_c|\theta^*)}{\pi(\theta^*)q(\theta^*|\theta_c)}\right\}$

where $\theta^*$ is the proposed value and $\theta_c$ is the current value (picked at random from the particle population), while q is a proposal kernel used to simulate the proposed value. The algorithm is adaptive in that the previous population of particles is used to make the choice of the proposal q, as well as of the tolerance level $\epsilon_t$. Although the method is valid as a particle system applied in the ABC setting, I have difficulties to gauge the level of novelty of the method (then applied to a model of Riley et al., 2003, J. Theoretical Biology). Learning from previous particle populations to build a better kernel q is indeed a constant feature in SMC methods, from Sisson et al.’s ABC-PRC (2007)—note that Drovandi and Pettitt mistakenly believe the ABC-PRC method to include partial rejection control, as argued in this earlier post—, to Beaumont et al.’s ABC-PMC (2009). The paper also advances the idea of adapting the tolerance on-line as an $\alpha$ quantile of the previous particle population, but this is the same idea as in Del Moral et al.’s ABC-SMC. The only strong methodological difference, as far as I can tell, is that the MCMC steps are repeated “numerous times” in the current paper, instead of once as in the earlier papers. This however partly cancels the appeal of an O(N) order method versus the O() order PMC and SMC methods. An interesting remark made in the paper is that more advances are needed in cases when simulating the pseudo-observations is highly costly, as in Ising models. However, replacing exact simulation [as we did in the model choice paper] with a Gibbs sampler cannot be that detrimental.

### 4 Responses to “Another ABC paper”

1. […] done in an adaptive way, while it could, given the recent 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 […]

2. […] algorithm of Pierre Del Moral, Arnaud Doucet and Ajay Jasra [not referenced in the current paper], rhe similar algorithm of Chris Drovandi and Tony Pettitt, and our recent paper with  Jean-Michel Marin, Pierre Pudlo and […]

3. Hi, Scott! I understand your annoyment about my quibbling, but, from a mathematical point of view, the PRC in ABC-PRC is a misnomer: I can only repeat my earlier argument, namely that there is no non-zero constant such that it is smaller than an arbitrary series of positive real numbers. (This is the argument “Suppose we then implement the PRC algorithm for some c > 0 such that only identically zero weights are smaller than c” in the Appendix of the original paper.) Your argument [here] of using the machine tolerance is not a mathematical argument, but rather a computer trick. I do not see any deep consequence in the fact that ABC-PRC is not PRC, though, since the analysis of the method is done from a standard importance sampling perspective…. Cheers, X.

4. Dear Christian,

I’m really not sure why you continue to maintain that ABC-PRC does not contain PRC.

Setting the PRC threshold, c, equal to machine tolerance would seem to “solve” your problem. In this case c>0 is satisfied, and only exactly zero weights are less than c.

All the best,

Scott

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