[FAQ] Homework 4 (2026): Why does the maximum F1 threshold (0.41) not match the Precision-Recall intersection (0.63)? #486

Course

machine-learning-zoomcamp

Question

In Homework 4, Question 3 identifies the decision threshold where Precision and Recall intersect (0.63), while Question 4 finds the threshold that maximizes the F1-score (0.41).

Why does the maximum F1-score not occur at the intersection where Precision equals Recall?

Answer

A common intuition is that the harmonic mean ($F_1$) is always highest where precision and recall are equal ($P \approx R$). In practice, this is rarely true, especially on imbalanced datasets or models with asymmetric score distributions.

The $F_1$-score formula is:
$$F_1 = 2 \cdot \frac{P \cdot R}{P + R}$$

Comparing the two thresholds on the validation set:

1. Precision–Recall intersection (threshold = 0.63):

   • $Precision = 0.733$, $Recall = 0.725$
   • Here, the costs of False Positives and False Negatives are balanced ($|P - R| \approx 0.0075$).
   • Resulting $F_1 \approx 0.729$.

2. Maximum $F_1$ threshold (threshold = 0.41):

   • Lowering the threshold to 0.41 boosts Recall to 0.976 while Precision drops to 0.619.
   • Because $F_1$ is driven by the product $P \cdot R$, the $+25.1%$ gain in recall significantly outweighs the $-11.4%$ loss in precision.
   • Resulting $F_1 = \mathbf{0.758}$ (the global maximum).

Conclusion:
Balancing Precision and Recall ($P \approx R$) minimizes the gap between metrics, but does not necessarily maximize the harmonic mean ($F_1$). The optimal $F_1$ threshold frequently favors recall when capturing positive cases yields a net gain in the harmonic mean.

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