Sufficient statistics for ABC
The issue of picking quasi-sufficient statistics in ABC algorithms is quite important and Joyce and Marjoram have published (last August, sorry I’m late!) a proposal in Statistical Applications in Genetics and Molecular Biology that is addressing this issue, at least to the extend that the inclusion of a new statistics within the set of summary statistics can be assessed in terms of likelihood ratio. There are difficulties with the method, first and foremost that the construction of the statistics is not discussed, while the method is not independent from parameterisation, and also that the order in which the statistics are considered is determinant for their inclusion/exclusion. A regularisation of the method proposed at the end of the paper is to use a forward-backward selection mechanism. This does not address another issue, namely the impact of the correlation between the summary statistics. Note at last that the method still depends on an approximation factor that needs to be determined by the statistician/experimenter running the algorithm.