Relevant statistics for Bayesian model choice

The choice of the summary statistics in Bayesian inference and in particular in ABC algorithms is paramount to produce a valid outcome. We derive necessary and sufficient conditions on those statistics for the corresponding Bayes factor to be convergent, namely to asymptotically select the true model. Those conditions which amount to the means of the summary statistics to asymptotically differ under both models are then usable in ABC settings to determine which summary statistics are appropriate, most generally via a standard Monte Carlo validation.