## Reply to our PNAS letter

In a well-coordinated move, Oliver Ratmann and his co-authors published his reply to our letter on arXiv yesterday afternoon, right after the “viva” for his PhD thesis at Imperial. The arXiv reply is the longer version of the PNAS reply (just like our arXiv posting is the longer version of our letter). Maybe the most important feature of the PNAS paper is that the original motivation of running ABC for conducting inference on the parameters of a model is replaced by the alternative goal of running ABC for assessing a model from within, the parameters being then at best nuisance parameters. While unwilling to restart the debate, I (obviously) still stick to the arguments presented in our letter, in particular that what serves as a proxy to a likelihood in ABCμ is not always a density in the observed variable $x_0$ if only because the use of summary statistics induces a decrease in the dimension of the vector. (A note about the Poisson example: when we mentioned the “truncation to positive values”, we meant that, since $x_0$ is a positive integer valued random variable, its density given $\epsilon$ should only be positive for positive integer values if it is to be interpreted as a density in $x_0$.)  Maybe my strongest misgiving about using ABCμ for model criticism is that I do not see any reason why the discrepancies $\epsilon_k$ should be centred at zero given the data $x_0$… In a frequentist approach $\epsilon=x-x_0$ is indeed symmetric and thus centred at zero (if the model is correct) but from a Bayesian perspective this is not the case. (Hence my earlier criticism of the null hypothesis.) It would thus be nice to see the equivalent of Example 3 and Figures 1-2 when the data is truly from the tested model: zero could then be (more) covered by the posterior of $\epsilon$ but there is no reason this posterior should be centred there! Anyway, I find this reply quite stimulating for  further pursuing the potential of ABC methods in setups where there is no viable alternatives.

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