## Another ABC paper

Posted in Statistics with tags , , , , , , , on July 24, 2010 by xi'an

“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.

## Inference in epidemic models w/o likelihoods

Posted in Statistics with tags , , , , , , , , , on July 21, 2010 by xi'an

“We discuss situations in which we think simulation-based inference may be preferable to likelihood-based inference.” McKinley, Cook, and Deardon, IJB

I only became aware last week of the paper Inference in epidemic models without likelihoods by McKinley, Cook and Deardon, published in the International Journal of Biostatistics in 2009. (Anyone can access the paper by becoming a guest, ie providing some information.) The paper is essentially a simulation experiment comparing ABC-MCMC and ABC-SMC with regular data augmentation MCMC. The authors experiment on the tolerance level, the choice of metric and of summary statistics, in an exponential inter-event process modelling. The setting is interesting, in particular because it applies to highly dangerous diseases like the Ebola fever—for which there is no known treatment and which is responsible for a 88% decline in observed chimpanzee populations since 2003!—. The conclusions are overall not highly surprising, namely that repeating simulations of the data points given one simulated parameter does not seem to contribute [much] to an improved approximation of the posterior by the ABC sample, that the tolerance level does not seem to be highly influential, that the choice of the summary statistics and of the calibration factors are important, and that ABC-SMC outperforms ABC-MCMC (MCMC remaining the reference). Slightly more surprising is the conclusion that the choice of the distance/metric influences the outcome. (I failed to read in the paper strong arguments supporting the above sentence stolen from the abstract.)

“There are always doubts that the estimated posterior really does correspond to the true posterior.” McKinley, Cook, and Deardon, IJB

On the “negative” side, this paper is missing the recent literature both on the nonparametric aspects of ABC and on the more adaptive [PMC] features of ABC-SMC, as processed in our Biometrika ABC-PMC paper and in Del Moral, Doucet and Jasra. (Again, this is not a criticism in that the paper got published in early 2009.) I think that using past simulations to build the proposal and the next tolerance and, why not!, the relevant statistics, would further the improvement brought by sequential methods. (The authors were aware of the correction of Sisson et al., and used instead the version of Toni et al. They also mention the arXived notes of Marc Beaumont, which started our prodding into ABC-PRC.) The comparison experiment is based on a single dataset, with fixed random walk variances for the MCMC algorithms, while the prior used in the simulation seems to me to be highly peaked around the true value (gamma rates of 0.1). Some of the ABC scenari do produce estimates that are rather far away from the references given by MCMC, take for instance the CABC-MCMC when the tolerance ε is 10 and R is 100.