What are TPR and FPR? How are they related to precision and recall?

TPR (True Positive Rate) and FPR (False Positive Rate) are metrics used to evaluate the quality of binary classification.

• TPR (sensitivity, recall) — the proportion of correctly predicted positive cases out of all actual positive cases:

  [ TPR = \frac{TP}{TP + FN} ]

• FPR — the proportion of incorrectly predicted positive cases out of all actual negative cases:

  [ FPR = \frac{FP}{FP + TN} ]

Here, TP — true positives, FP — false positives, FN — false negatives, TN — true negatives.

Precision — the proportion of correctly predicted positive cases out of all predicted positive cases:

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

Relationship between them:

• Recall and TPR are synonyms.
• Precision and recall together help understand the balance between completeness and accuracy of the classification.

Example: If the model often incorrectly classifies negative objects as positive (high FPR), precision drops because of many FP. If the model misses many positives (low TPR/recall), then completeness suffers.