ABC model choice

I was re-reading the recently arXived paper by Toni and Stumpf on ABC based model choice and, besides noticing that their Gibbs random field example (§3.2) is the same as ours (§3.1), down to the prior choice, this led me to wonder about the choice of the ABC distance in those settings. On the one hand, the statistical perspective is to compare the predictive performances of different models and hence use the same distance for all models. On the other hand, the ABC perspective implies using different summary statistics for different models, hence using different distances… In a “true” model there is no issue because we end up comparing (margina) likelihoods but in an approximation like ABC, given that we replace the data with a summary statistic, then the distribution of a summary statistic with an indicator of proximity, we end up with paradoxes like this, where we compare pseudo-distributions of objects of different dimensions. (Toni and Stumpf made the choice in their paper of pulling all summary statistics together into an overall distance, while in ours we had the special Gibbs property of a sufficient statistic across models…)