Precision and recall are both metrics

$$\text{Precision} = \frac{TP}{TP + FP} $$

A high precision indicates that the model has a low rate of false positives, meaning that when it predicts a positive outcome, it is usually correct This metric is especially important in contexts where false positives are costly, such as spam detection, where incorrectly marking a legitimate email as spam can be more detrimental than missing a spam email

Recall, also known as sensitivity or the true positive rate, measures the model’s ability to identify all actual positive instances It is defined as the ratio of true positives to the sum of true positives and false negatives (FN), given by the formula:

$$\text{Recall} = \frac{TP}{TP + FN} $$

A high recall means that the model successfully captures most of the relevant cases, minimizing the number of missed positives This is critical in safety-critical applications such as medical diagnostics or weapon detection, where failing to identify a real threat (a false negative) is far more dangerous than issuing a false alarm (a false positive)

There is often a trade-off between precision and recall: increasing one typically results in a decrease in the other For example, lowering the classification threshold to capture more positive cases improves recall but may also increase false positives, thereby reducing precision. Conversely, raising the threshold enhances precision by reducing false positives but risks missing actual positives, lowering recall

Both metrics are essential for a comprehensive evaluation of model performance, and they are frequently combined into a single measure called the F-score (or F1-score), which is the harmonic mean of precision and recall:

$$F1 = 2 \times \frac{\text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}} $$

The F1-score provides a balanced assessment, especially when both precision and recall are important While perfect precision and recall (both equal to 1) represent an ideal model, real-world systems often require balancing these metrics based on the specific application’s priorities