## approximate approximate Bayesian computation [not a typo!]

“Our approach in handling the model uncertainty has some resemblance to statistical ‘‘emulators’’ (Kennedy and O’Hagan, 2001), approximative methods used to express the model uncertainty when simulating data under a mechanistic model is computationally intensive. However, emulators are often motivated in the context of Gaussian processes, where the uncertainty in the model space can be reasonably well modeled by a normal distribution.” Pierre Pudlo pointed out to me the paper AABC: Approximate approximate Bayesian computation for inference in population-genetic models by Buzbas and Rosenberg that just appeared in the first 2015 issue of Theoretical Population Biology. Despite the claim made above, including a confusion on the nature of Gaussian processes, I am rather reserved about the appeal of this AA rated ABC…

“When likelihood functions are computationally intractable, likelihood-based inference is a challenging problem that has received considerable attention in the literature (Robert and Casella, 2004).”

The ABC approach suggested therein is doubly approximate in that simulation from the sampling distribution is replaced with simulation from a substitute cheaper model. After a learning stage using the costly sampling distribution. While there is convergence of the approximation to the genuine ABC posterior under infinite sample and Monte Carlo sample sizes, there is no correction for this approximation and I am puzzled by its construction. It seems (see p.34) that the cheaper model is build by a sort of weighted bootstrap: given a parameter simulated from the prior, weights based on its distance to a reference table are constructed and then used to create a pseudo-sample by weighted sampling from the original pseudo-samples. Rather than using a continuous kernel centred on those original pseudo-samples, as would be the suggestion for a non-parametric regression. Each pseudo-sample is accepted only when a distance between the summary statistics is small enough. This bootstrap flavour is counter-intuitive in that it requires a large enough sample from the true  sampling distribution to operate with some confidence… I also wonder at what happens when the data is not iid.  (I added the quote above as another source of puzzlement, since the book is about cases when the likelihood is manageable.)

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