Micro AUROC

1. Binary AUROC Recap

• AUROC (Area Under ROC Curve) = probability that the model ranks a randomly chosen positive sample higher than a randomly chosen negative sample.
• Well-defined for binary classification.

---

2. Multiclass Extension

When we have K classes, AUROC needs to be extended. Two common strategies are:

• Macro AUROC: average AUROC for each class (OvR).
• Micro AUROC: aggregate all predictions across classes, then compute AUROC globally.

---

3. Definition of Micro AUROC

• Flatten all class predictions into a single pool of binary decisions (one-vs-rest, across all classes).
• Then compute TP, FP, TN, FN globally across all classes.
• Finally, calculate AUROC as if it were a single binary task.

Formally:

$AUROC_{micro} = AUROC\left(\sum_{i=1}^{K} \text{positives for class } i, \; \sum_{i=1}^{K} \text{negatives for class } i \right)$

---

4. Key Characteristics

• Micro AUROC = sample-level discrimination ability.
• Large (majority) classes dominate, because counts are pooled.
• If your dataset is imbalanced, Micro AUROC may look high even if rare classes perform poorly.
• Complements Macro AUROC, which treats all classes equally.

---

5. Example

Suppose we have 3 classes (A, B, C):

• AUROC(A vs rest) = 0.90
• AUROC(B vs rest) = 0.70
• AUROC(C vs rest) = 0.60
• Macro AUROC = (0.90 + 0.70 + 0.60) / 3 = 0.733
• Micro AUROC is computed by pooling all predictions: if class A has many more samples, Micro AUROC will be closer to 0.90 (dominated by majority class).

---

6. Interpretation

• Micro AUROC: best when you want to evaluate the overall ability of the classifier across all samples.
• Macro AUROC: best when you want fairness across classes, especially minority ones.

---

Summary

• Micro AUROC = AUROC computed after pooling predictions across all classes.
• Dominated by majority classes.
• Useful for overall performance measurement, but hides minority class weaknesses.
• Always report both Macro and Micro AUROC for a balanced evaluation.