Average Precision

Average Precision (AP) is the area under the precision-recall curve — measuring ranking quality by averaging precision at each relevant result position, capturing both precision and recall in a single metric.

What Is Average Precision?

• Definition: Average of precision values at positions where relevant items appear.
• Formula: AP = (Σ P(k) × rel(k)) / (total relevant items).
• Range: 0 (worst) to 1 (perfect).

How AP Works

1. Rank items by predicted relevance.
2. For each relevant item at position k, compute Precision@k.
3. Average these precision values.

Example

Ranked list: R, N, R, R, N (R=relevant, N=not relevant).
- P@1 = 1/1 = 1.0 (1st relevant at position 1).
- P@3 = 2/3 = 0.67 (2nd relevant at position 3).
- P@4 = 3/4 = 0.75 (3rd relevant at position 4).
- AP = (1.0 + 0.67 + 0.75) / 3 = 0.81.

Why Average Precision?

• Position-Aware: Rewards relevant items at top positions.
• Comprehensive: Considers all relevant items, not just top-K.
• Single Metric: Combines precision and recall.
• Ranking Quality: Measures overall ranking effectiveness.

AP vs. Other Metrics

vs. Precision@K: AP considers all positions, P@K only top-K.
vs. NDCG: AP binary relevance, NDCG handles graded relevance.
vs. MRR: AP considers all relevant items, MRR only first.

Applications: Information retrieval, search evaluation, recommendation evaluation, object detection (mAP).

Tools: scikit-learn, IR evaluation libraries.

Average Precision is comprehensive ranking evaluation — by averaging precision at all relevant positions, AP captures both the quality and completeness of rankings in a single, interpretable metric.