Maybe it is just a coincidence, but both most recent issues of Bayesian Analysis have an article featuring approximate Bayesian inference. One is by Daniel Add Contact Form Graham and co-authors on Approximate Bayesian Inference for Doubly Robust Estimation, while the other one is by Chris Drovandi and co-authors from QUT on Exact and Approximate Bayesian Inference for Low Integer-Valued Time Series Models with Intractable Likelihoods. The first paper has little connection with ABC. Even though it (a) uses a lot of three letter acronyms [which does not help with speed reading] and (b) relies on moment based and propensity score models. Instead, it relies on Bayesian bootstrap, which suddenly seems to me to be rather connected with empirical likelihood! Except the weights are estimated via a Dirichlet prior instead of being optimised. The approximation lies in using the bootstrap to derive a posterior predictive. I did not spot any assessment or control of the approximation effect in the paper.
The second paper connects pMCMC with ABC. Plus pseudo-marginals on the side! And even simplified reversible jump MCMC!!! I am far from certain I got every point of the paper, though, especially the notion of dimension reduction associated with this version of reversible jump MCMC. It may mean that latent variables are integrated out in approximate (marginalised) likelihoods [as explicated in Andrieu and Roberts (2009)].
“The difference with the common ABC approach is that we match on observations one-at-a-time” (p.328)
The model that the authors study is an integer value time-series, like the INAR(p) model. Which integer support allows for a non-zero probability of exact matching between simulated and observed data. One-at-a-time as indicated in the above quote. And integer valued tolerances like ε=1 otherwise. In the case auxiliary variables are necessary, the authors resort to the alive particle filter of Jasra et al. (2013), which main point is to produce an unbiased estimate of the (possibly approximate) likelihood, to be exploited by pseudo-marginal techniques. However, unbiasedness sounds less compelling when moving to approximate methods, as illustrated by the subsequent suggestion to use a more stable estimate of the log-likelihood. In fact, when the tolerance ε is positive, the pMCMC acceptance probability looks quite close to an ABC-MCMC probability when relying on several pseudo-data simulations. Which is unbiased for the “right” approximate target. A fact that may actually holds for all ABC algorithms. One quite interesting aspect of the paper is its reflection about the advantage of pseudo-marginal techniques for RJMCMC algorithms since they allow for trans-dimension moves to be simplified, as they consider marginals on the space of interest. Up to this day, I had not realised Andrieu and Roberts (2009) had a section on this aspect… I am still unclear about the derivation of the posterior probabilities of the models under comparison, unless it is a byproduct of the RJMCMC algorithm. A last point is that, for some of the Markov models used in the paper, the pseudo observations can be produced as a random one-time move away from the current true observation, which makes life much easier for ABC and explain why exact simulations can sometimes be produced. (A side note: the authors mention on p.326 that EP is only applicable when the posterior is from an exponential family, while my understanding is that it uses an exponential family to approximate the true posterior.)