The Effect of Class Imbalance on Precision-Recall Curves

In this note I study how the precision of a classifier depends on the ratio $r$ of positive to negative cases in the test set, as well as the classifiers true and false positive rates. This relationship allows prediction of how the precision-recall curve will change with $r$, which seems not to be well known. It also allows prediction of how $F_{beta}$ and the Precision Gain and Recall Gain measures of Flach and Kull (2015) vary with $r$.