improving synthetic likelihood

Chris Drovandi gave an after-dinner [QUT time!] talk for the One World ABC webinar on a recent paper he wrote with Jacob Proddle, Scott Sisson and David Frazier. Using a regular MCMC step on a synthetic likelihood approximation to the posterior. Or a (simulation based) unbiased estimator of it.

By evaluating the variance of the log-likelihood estimator, the authors show that the number of simulations n need scale like n²d² to keep the variance under control. And suggest PCA decorrelation of the summary statistic components as a mean to reduce the variance since it then scales as n²d. Rather idly, I wonder at the final relevance of precisely estimating the (synthetic) likelihood when considering it is not the true likelihood and when the n² part seems more damning. Moving from d² to d seems directly related to the estimation of a full correlation matrix for the Normal synthetic distribution of the summary statistic versus the estimation of a diagonal matrix. The usual complaint that performances highly depend on the choice of the summary statistic also applies here, in particular when its dimension is much larger than the dimension d of the parameter (as in the MA example). Although this does not seem to impact the scale of the variance.

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.