# Consistent ranked list for ROC interpolation

For classifiers with binary outputs, their performance is summarized by a true positive rate and false positive rate. To interpolate the performance between two classifiers $A$ and $B$ with their respective (true positive rate, false positive rate) being $(t_A, f_A)$, and $(t_B,f_B)$, we can construct a new classifier to achieve true positive rate of $t_A+(t_A-t_B)\frac{f_C-f_B}{f_A-f_B}$ at a false positive rate of $f_C$ by sampling. For any data point $x$, take the label $\hat{y}_C$ from the classifier $A$ (i.e., $\hat{y}_C \leftarrow \hat{y}_A$) with the probability $p_A = \frac{f_C-f_B}{f_A-f_B}$, and from $B$ (i.e., $\hat{y}_C \leftarrow \hat{y}_B$) with probability $1-p_A$.

My question is, can we interpolate between at points so we achieve a ranked list? i.e., assuming $f_B<f_A$, as we go from $f_B$ to $f_A$, we keep adding to the list of points predicted to be Class 1?

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