consistency of ABC
Along with David Frazier and Gael Martin from Monash University, Melbourne, we have just completed (and arXived) a paper on the (Bayesian) consistency of ABC methods, producing sufficient conditions on the summary statistics to ensure consistency of the ABC posterior. Consistency in the sense of the prior concentrating at the true value of the parameter when the sample size and the inverse tolerance (intolerance?!) go to infinity. The conditions are essentially that the summary statistics concentrates around its mean and that this mean identifies the parameter. They are thus weaker conditions than those found earlier consistency results where the authors considered convergence to the genuine posterior distribution (given the summary), as for instance in Biau et al. (2014) or Li and Fearnhead (2015). We do not require here a specific rate of decrease to zero for the tolerance ε. But still they do not hold all the time, as shown for the MA(2) example and its first two autocorrelation summaries, example we started using in the Marin et al. (2011) survey. We further propose a consistency assessment based on the main consistency theorem, namely that the ABC-based estimates of the marginal posterior densities for the parameters should vary little when adding extra components to the summary statistic, densities estimated from simulated data. And that the mean of the resulting summary statistic is indeed one-to-one. This may sound somewhat similar to the stepwise search algorithm of Joyce and Marjoram (2008), but those authors aim at obtaining a vector of summary statistics that is as informative as possible. We also examine the consistency conditions when using an auxiliary model as in indirect inference. For instance, when using an AR(2) auxiliary model for estimating an MA(2) model. And ODEs.
Xi'an's Og 
September 2, 2015 at 6:36 pm
I enjoyed reading this, and think it makes a nice link to the indirect inference work on binding functions.
I have a comment on the example in Section 6 using distance (13). It seems to me that this shows a lack of consistency under lim eps->0 lim T->infty. However for any fixed finite T taking eps->0 recovers the true posterior. So I think this example would meet an alternative definition of consistency where the order of the limits is switched.
August 25, 2015 at 8:59 am
I haven’t finished reading this yet, but so far it feels more like a first step or case study of ABC consistency: in the sense that the case of R^n summary statistics and the rejection ABC (indicator function) kernel is somewhat vanilla compared to e.g. an infinite-dimensional summary statistic (e.g. Weyant et al., 2013) with generic kernel of bandwidth epsilon and various other flavours of ABC?
Similarly, I would have thought a lot of summary statistics would converge to their limiting value almost surely, so in that case S1 could be weakened allowing S2 to be tightened, etc.