## delayed acceptance ABC-SMC

Posted in pictures, Statistics, Travel with tags , , , , , , , on December 11, 2017 by xi'an

Last summer, during my vacation on Skye,  Richard Everitt and Paulina Rowińska arXived a paper on delayed acceptance associated with ABC. ArXival that I missed, then! In order to decrease the number of simulations from the likelihood. As in our own delayed acceptance paper (without ABC), a cheap alternative generator is used to first reject the least likely parameters values, before possibly continuing to use a full generator. Also as lazy ABC. The first step of this ABC algorithm requires a cheap generator plus a primary tolerance ε¹ to compare the generation with the data or part of it. This may be followed by a second generation with a second tolerance level ε². The paper applies more specifically ABC-SMC as introduced in Sisson, Fan and Tanaka (2007) and reassessed in our subsequent 2009 Biometrika paper with Mark Beaumont, Jean-Marie Cornuet and Jean-Michel Marin. As well as in the ABC-SMC paper by Pierre Del Moral and Arnaud Doucet.

When looking at the version of the algorithm [Algorithm 2] based on two basic acceptance ABC steps, there are two features I find intriguing: (i) the primary step uses a cheap generator to reject early poor values of the parameter, followed by the second step involving a more expensive and exact generator, but I see no impact of the choice of this cheap generator in the acceptance probability; (ii) this is an SMC algorithm with imposed resampling at each iteration but there is no visible step for creating new weights after the resampling step. In the current presentation, it sounds like the weights do not change from the initial step, except for those turning to zero and the renormalisation transforms. Which makes the (unspecified) stratification of little interest if any. I must therefore miss a point in the implementation!

One puzzling sentence in the appendix is that the resampling algorithm used in the SMC step “ensures that every particle that is alive before resampling is represented in the resampled particles”, which reminds me of an argument [possibly a different one] made already in Sisson, Fan and Tanaka (2007) and that we could not validate in our subsequent paper. For resampling to be correct, a form of multinomial sampling must be implemented, even via variance reduction schemes like stratified or systematic sampling.

## lazy ABC…what?!

Posted in Kids, pictures, Statistics with tags , , , , , on November 8, 2017 by xi'an

## lazy ABC

Posted in Books, Statistics, University life with tags , , , , , , , on June 9, 2014 by xi'an

“A more automated approach would be useful for lazy versions of ABC SMC algorithms.”

Dennis Prangle just arXived the work on lazy ABC he had presented in Oxford at the i-like workshop a few weeks ago. The idea behind the paper is to cut down massively on the generation of pseudo-samples that are “too far” from the observed sample. This is formalised through a stopping rule that puts the estimated likelihood to zero with a probability 1-α(θ,x) and otherwise divide the original ABC estimate by α(θ,x). Which makes the modification unbiased when compared with basic ABC. The efficiency appears when α(θ,x) can be computed much faster than producing the entire pseudo-sample and its distance to the observed sample. When considering an approximation to the asymptotic variance of this modification, Dennis derives a optimal (in the sense of the effective sample size) if formal version of the acceptance probability α(θ,x), conditional on the choice of a “decision statistic” φ(θ,x).  And of an importance function g(θ). (I do not get his Remark 1 about the case when π(θ)/g(θ) only depends on φ(θ,x), since the later also depends on x. Unless one considers a multivariate φ which contains π(θ)/g(θ) itself as a component.) This approach requires to estimate

$\mathbb{P}(d(S(Y),S(y^o))<\epsilon|\varphi)$

as a function of φ: I would have thought (non-parametric) logistic regression a good candidate towards this estimation, but Dennis is rather critical of this solution.

I added the quote above as I find it somewhat ironical: at this stage, to enjoy laziness, the algorithm has first to go through a massive calibration stage, from the selection of the subsample [to be simulated before computing the acceptance probability α(θ,x)] to the construction of the (somewhat mysterious) decision statistic φ(θ,x) to the estimation of the terms composing the optimal α(θ,x). The most natural choice of φ(θ,x) seems to be involving subsampling, still with a wide range of possibilities and ensuing efficiencies. (The choice found in the application is somehow anticlimactic in this respect.) In most ABC applications, I would suggest using a quick & dirty approximation of the distribution of the summary statistic.

A slight point of perplexity about this “lazy” proposal, namely the static role of ε, which is impractical because not set in stone… As discussed several times here, the tolerance is a function of many factors incl. all the calibration parameters of the lazy ABC, rather than an absolute quantity. The paper is rather terse on this issue (see Section 4.2.2). It seems to me that playing with a large collection of tolerances may be too costly in this setting.