In their article of 2006, The Relationship Between Precision-Recall and ROC Curves, J. Davis & M. Goadrich show that ROC (TPR vs FPR) and Precision-Recall (PR) curves have one-to-one correspondence. In particular, the authors prove equivalence of a classifier dominating another in either space (for a given mix of classes, A dominates B in ROC space <=> A dominates B in PR space). Now, the ROC curves are not affected by changing the mix of positive and negative classes, yet the PR curves are. Therefore, the equivalence is really not there -- because ROC dominance (of A over B) implies PR dominance for any mix (since the ROC curve remains the same) whereas PR dominance implies ROC dominance for a specific mix only. However, for a different mix it's conceivable that B may dominate A in the PR space. Is there some kind of a paradox here?