  Everybody writes that ROC curves, a very common tool in binary classification
problems, should be optimal, and in particular concave, non-decreasing and
above the 45-degree line. Everybody uses ROC curves, theoretical and especially
empirical, which are not so. This work is an attempt to correct this
schizophrenic behavior. Optimality stems from the Neyman-Pearson lemma, which
prescribes using likelihood-ratio based ROC curves. Starting from there, we
give the most general definition of a likelihood-ratio based classification
procedure, which encompasses finite, continuous and even more complex data
types. We point out a strict relationship with a general notion of
concentration of two probability measures. We give some nontrivial examples of
situations with non-monotone and non-continuous likelihood ratios. Finally, we
propose the ROC curve of a likelihood ratio based Gaussian kernel flexible
Bayes classifier as a proper default alternative to the usual empirical ROC
curve.
