David Nott, Chris Drovandi and Robert Kohn just arXived a paper on a comparison between ABC and synthetic likelihood, which is both interesting and timely given that synthetic likelihood seems to be lacking behind in terms of theoretical evaluation. I am however as puzzled by the results therein as I was by the earlier paper by Price et al. on the same topic. Maybe due to the Cambodia jetlag, which is where and when I read the paper.
My puzzlement, thus, comes from the difficulty in comparing both approaches on a strictly common ground. The paper first establishes convergence and asymptotic normality for synthetic likelihood, based on the 2003 MCMC paper of Chernozukov and Hong [which I never studied in details but appears like the MCMC reference in econometrics literature]. The results are similar to recent ABC convergence results, unsurprisingly when assuming a CLT on the summary statistic vector. One additional dimension of the paper is to consider convergence for a misspecified covariance matrix in the synthetic likelihood [and it will come back with a revenge]. And asymptotic normality of the synthetic score function. Which is obviously unavailable in intractable models.
The first point I am missing is why the computing time required for approximating mean and variance in the synthetic likelihood, by Monte Carlo means, is not accounted for in the comparison between ABC and synthetic likelihood versions. Remember that ABC only requires one (or at most two) pseudo-samples per parameter simulation. The latter requires M, which is later constrained to increase to infinity with the sample size. Simulations that are usually the costliest in the algorithms. If ABC were to use M simulated samples as well, since it already relies on a kernel, it could as well construct [at least on principle] a similar estimator of the [summary statistic] density. Or else produce M times more pairs (parameter x pseudo-sample).
Another point deals with the use of multiple approximate posteriors in the comparison. Since the approximations differ, it is unclear that convergence to a given approximation is all that should matter, if the approximation is less efficient [when compared with the original and out-of-reach posterior distribution]. Especially for a finite sample size n. This chasm in the targets becomes more evident when the authors discuss the use of a constrained synthetic likelihood covariance matrix towards requiring less pseudo-samples, i.e. lower values of M, because of a smaller number of parameters to estimate. This should be balanced against the loss in concentration of the synthetic approximation, as exemplified by the realistic examples in the paper. (It is also hard to see why M could be not of order √n for Monte Carlo reasons.)
The last section in the paper is revolving around diverse issues for misspecified models, from wrong covariance matrix to wrong generating model. As we just submitted a paper on ABC for misspecified models, I will not engage into a debate on this point but find the proposed strategy that goes through an approximation of the log-likelihood surface by a Gaussian process and a derivation of the covariance matrix of the score function apparently greedy in both calibration and computing. And not so clearly validated when the generating model is misspecified.