One-vs-Rest (OvR) AUROC

1. Binary AUROC Recap

• AUROC = Probability that a randomly chosen positive sample has a higher predicted score than a randomly chosen negative sample.
• Well-defined only for binary classification.

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2. Multiclass Case

When we have K classes, AUROC is not directly defined.

• Solution: Extend the binary idea using One-vs-Rest (OvR).
• For each class $i$:
  • Treat class i = positive.
  • Treat all other classes = negative.
  • Compute AUROCi_ii​ for this binary problem.

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3. Definition

$AUROC_{OvR}(i) = AUROC(\text{class}_i \; vs \; rest)$

Then you can:

• Report AUROC$_i$​ for each class separately.
• Or average them to get Macro AUROC:

$AUROC_{macro} = \frac{1}{K} \sum_{i=1}^K AUROC_{OvR}(i)$

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4. Example

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

• Compute OvR AUROC:
  • AUROC(A vs B+C) = 0.83
  • AUROC(B vs A+C) = 0.76
  • AUROC(C vs A+B) = 0.70
• Each AUROC$_i$​ tells you how well the model separates that class from all others.
• Macro AUROC = (0.83 + 0.76 + 0.70) / 3 = 0.763

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5. Interpretation

• One-vs-Rest AUROC gives a class-specific discrimination measure.
• Useful when you want to know: “How well does the model detect THIS class compared to the rest?”
• Especially important in imbalanced data, e.g., fraud detection (fraud = rare class).

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Summary

• One-vs-Rest AUROC = AUROC computed for one class vs all others.
• Extends AUROC from binary → multiclass.
• You can report per-class values or average (macro AUROC).
• Tells you how well each class is separable from the rest.