uncertainty in the ABC posterior

In the most recent Bayesian Analysis, Marko Järvenpää et al. (including my coauthor Aki Vehtari) consider an ABC setting where the number of available simulations of pseudo-samples  is limited. And where they want to quantify the amount of uncertainty resulting from the estimation of the ABC posterior density. Which is a version of the Monte Carlo error in practical ABC, in that this is the difference between the ABC posterior density for a given choice of summaries and a given choice of tolerance, and the actual approximation based on a finite number of simulations from the prior predictive. As in earlier works by Michael Gutmann and co-authors, the focus stands in designing a sequential strategy to decide where to sample the next parameter value towards minimising a certain expected loss. And in adopting a Gaussian process modelling for the discrepancy between observed data and simulated data, hence generalising the synthetic likelihood approach. This allows them to compute the expectation and the variance of the unnormalised ABC posterior, based on plugged-in estimators. From where the authors derive a loss as the expected variance of the acceptance probability (although it is not parameterisation invariant). I am unsure I see the point for this choice in that there is no clear reason for the resulting sequence of parameter choices to explore the support of the posterior distribution in a relatively exhaustive manner. The paper also mentions alternatives where the next parameter is chosen at the location where “the uncertainty of the unnormalised ABC posterior is highest”. Which sounds more pertinent to me. And further avoids integrating out the parameter. I also wonder if ABC mis-specification analysis could apply in this framework since the Gaussian process is most certainly a “wrong” model. (When concluding this post, I realised I had written a similar entry two years ago about the earlier version of the paper!)

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