Act 3 · Core metrics · 12 minutes

Two denominators. Two different questions.

Precision and recall use the same numerator—true positives—but look at two different groups.

Start from predictions

Precision

“When the model raises an alarm, how often is it right?”

\[\text{Precision}=\frac{TP}{TP+FP}\]

The denominator is everything predicted positive. False alarms reduce precision.

Prioritize precision when acting on a positive prediction is expensive: manual fraud reviews, blocking email, or invasive follow-up tests.

Start from reality

“Of all real positive cases, how many did the model catch?”

\[\text{Recall}=\frac{TP}{TP+FN}\]

The denominator is everything actually positive. Missed cases reduce recall.

Prioritize recall when missing a positive is dangerous: disease screening, fraud detection, or safety failures.

Predicted fraud

Predicted legitimate

Actual fraud

TP = 8caught

FN = 2missed

Actual legitimate

FP = 12false alarms

TN = 78cleared

\[\text{Precision}=\frac{8}{8+12}=40\%\]

\[\text{Recall}=\frac{8}{8+2}=80\%\]

Is this model better at catching fraud, or at making trustworthy fraud alerts?

It catches 80% of fraud, but only 40% of its alerts are genuine. It has stronger recall than precision.

The arithmetic mean can remain high even when one metric is weak. F1 uses the harmonic mean, which is pulled toward the smaller value.

\[F1=2\cdot\frac{\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{Recall}}\]

\[F1=2\cdot\frac{0.40\times0.80}{0.40+0.80}=0.53\]

F1 becomes high only when both precision and recall are reasonably high.

Scenario	Costlier mistake	Metric to emphasize
Medical screening	Missing a disease (FN)	Recall
Spam auto-delete	Deleting a real email (FP)	Precision
Fraud investigation queue	Both missed fraud and wasted reviews	F1 or cost-based metric

There is no universally best metric. The right metric reflects class balance, error cost, and what action follows a prediction.