Semi-automatic selection of summary statistics for ABC model choice

A central statistical goal is to choose between alternative explanatory models of data. In many modern applications, such as population genetics, it is not possible to apply standard methods based on evaluating the likelihood functions of the models, as these are numerically intractable. Approximate Bayesian computation (ABC) is a commonly used alternative for such situations. ABC simulates data x for many parameter values under each model, which is compared to the observed data x obs. More weight is placed on models under which S(x) is close to S(x obs), where S maps data to a vector of summary statistics. Previous work has shown the choice of S is crucial to the efficiency and accuracy of ABC. This paper provides a method to select good summary statistics for model choice. It uses a preliminary step, simulating many x values from all models and fitting regressions to this with the model as response. The resulting model weight estimators are used as S in an ABC analysis. Theoretical results are given to justify this as approximating low dimensional sufficient statistics. A substantive application is presented: choosing between competing coalescent models of demographic growth for Campylobacter jejuni in New Zealand using multi-locus sequence typing data.