Inference for stochastic simulation models by ABC
Hartig et al. published a while ago (2011) a paper in Ecology Letters entitled “Statistical inference for stochastic simulation models – theory and application”, which is mostly about ABC. (Florian Hartig pointed out the paper to me in a recent blog comment. about my discussion of the early parts of Guttman and Corander’s paper.) The paper is largely a tutorial and it reminds the reader about related methods like indirect inference and methods of moments. The authors also insist on presenting ABC as a particular case of likelihood approximation, whether non-parametric or parametric. Making connections with pseudo-likelihood and pseudo-marginal approaches. And including a discussion of the possible misfit of the assumed model, handled by an external error model. And also introducing the notion of informal likelihood (which could have been nicely linked with empirical likelihood). A last class of approximations presented therein is called rejection filters and reminds me very much of Ollie Ratman’s papers.
“Our general aim is to find sufficient statistics that are as close to minimal sufficiency as possible.” (p.819)
As in other ABC papers, and as often reported on this blog, I find the stress on sufficiency a wee bit too heavy as those models calling for approximation almost invariably do not allow for any form of useful sufficiency. Hence the mathematical statistics notion of sufficiency is mostly useless in such settings.
“A basic requirement is that the expectation value of the point-wise approximation of p(Sobs|φ) must be unbiased” (p.823)
As stated above the paper is mostly in tutorial mode, for instance explaining what MCMC and SMC methods are. As illustrated by the above figure. There is however a final and interesting discussion section on the impact of estimating the likelihood function at different values of the parameter. However, the authors seem to focus solely on pseudo-marginal results to validate this approximation, hence on unbiasedness, which does not work for most ABC approaches that I know. And for the approximations listed in the survey. Actually, it would be quite beneficial to devise a cheap tool to assess the bias or extra-variation due to the use of approximative techniques like ABC… A sort of 21st Century bootstrap?!
Xi'an's Og 
February 14, 2015 at 7:15 pm
Hartig et al. published paper presents a different perspective, but I think not quite direct as it should/could be.
It is why I always keep going back to Galton’s 1880 two stage qunincunx implementation of ABC. Here one can see a perhaps “hunter-gather” logic.
When as boy, I fished in a row boat. When the water was not too deep I could see schools of fish I did not want to catch (rock bass) but sometimes on the edge would be a fish I did want to catch (small mouth bass). I would row towards that fish and into deeper waters behind it, hoping to find their colleagues. Similarly, for a minnow (if I was after bait) towards the minnow into shallower water. Bayes theorem just abstracts and formalizes that logic.
Also, not to appear too much like a little boy with a hammer, Hartig et al. do seem to miss the continuity is only an approximation – “For continuous variables, the probability of observing exactly the same outcome is infinitesimally small.”
(No one, can by definition, confirm that the decimal places remain zero for all after k decimal places.)