ROC-AUC

sweeping every classification threshold, plot the true positive rate $TPR = \frac{TP}{TP+FN}$ (= recall) on the y-axis against the false positive rate $FPR = \frac{FP}{FP+TN}$ (= 1 - specificity) on the x-axis; AUC is the area under this curve. As the threshold drops, TPR rises monotonically while FPR grows too; key points: (0,0) = predict all negative, (1,1) = predict all positive, and the diagonal $TPR = FPR$ is random guessing (AUC = 0.5). Probabilistic interpretation: AUC = the probability that a randomly chosen positive scores higher than a randomly chosen negative, $AUC = P(s_{+} > s_{-})$ — threshold-free and class-ratio-free, a pure ranking metric. Hand computation: 4 samples with scores [0.9, 0.7, 0.5, 0.3] and labels [+, -, +, -]: there are $2 \times 2 = 4$ positive–negative pairs, and 3 of them (0.9, 0.7), (0.9, 0.3), (0.5, 0.3) are correctly ordered while only (0.5, 0.7) is inverted, so $AUC = 3/4 = 0.75$.

binary classification and ranking-quality evaluation (credit, recsys, search); interview staple “how is AUC computed and interpreted”; also used for model selection and tuning since it needs no fixed threshold.

accuracy and PR require a chosen threshold to compare models, while AUC decouples ranking quality from the threshold — one number summarizing all thresholds; however, under extreme imbalance the FPR denominator $FP+TN$ is inflated by the huge negative population, so a few false positives barely move FPR and the ROC curve looks over-optimistic — switch to the PR curve (sensitive to the $P = \rho$ baseline, exposing models whose ranking is polluted by negatives); AUC is class-ratio insensitive because it is rank-based, and it does not tell you the operating point — pair it with the threshold curve to pick TPR/FPR.

Hand-compute AUC: 4 samples with scores [0.9, 0.7, 0.5, 0.3] and labels [+, -, +, -], counting positive–negative pairs.

What is the probabilistic interpretation of AUC? Why is random guessing AUC ≈ 0.5 and a perfect ranking AUC = 1?

How to draw the ROC curve from model scores (threshold sweep updating TPR/FPR each step)? How is AUC computed by the trapezoid rule?

Why prefer the PR curve over ROC under extreme class imbalance, and why is AUC insensitive to the class ratio?

How to pick an operating threshold for a model with AUC = 0.75? Why doesn't AUC directly give a TPR/FPR working point?