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.

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