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In the realm of machine learning evaluation, sensitivity (also known as the true positive rate or hit rate) stands as one of the most critical metrics for assessing how well a classifier captures positive instances. This metric answers a fundamental question: "Of all the samples that are actually positive, what fraction did our model correctly identify?"
Sensitivity is formally defined as:
$$\text{Sensitivity} = \frac{TP}{TP + FN}$$
Where:
Intuitive Understanding: Imagine a medical screening test designed to detect a disease. A sensitivity of 0.95 means that out of every 100 people who actually have the disease, the test correctly identifies 95 of them. The remaining 5 are "missed" (false negatives). In high-stakes scenarios like cancer detection or fraud prevention, maximizing sensitivity is often crucial because missing a true positive can have severe consequences.
The Trade-off Consideration: While high sensitivity is desirable, it often comes at the cost of increased false positives. A model that predicts everything as positive would achieve perfect sensitivity (1.0) but would be practically useless. Understanding this trade-off is essential for building effective classifiers.
Your Task: Implement a function that computes the sensitivity score for a binary classification model. The function should:
actual_labels = [1, 0, 1, 1, 0, 1]
predicted_labels = [1, 0, 1, 0, 0, 1]0.75Let's identify the components:
• Total actual positives (1s in actual_labels): 4 instances (positions 0, 2, 3, 5) • True Positives: Positions where actual=1 AND predicted=1: positions 0, 2, 5 → 3 instances • False Negatives: Positions where actual=1 AND predicted=0: position 3 → 1 instance
Sensitivity = TP / (TP + FN) = 3 / (3 + 1) = 3/4 = 0.75
The model successfully identified 75% of all positive cases.
actual_labels = [1, 1, 1, 0, 0]
predicted_labels = [1, 1, 1, 0, 1]1.0Analyzing the classification:
• Total actual positives: 3 instances (positions 0, 1, 2) • True Positives: All 3 actual positives were correctly predicted → 3 instances • False Negatives: No positive samples were missed → 0 instances
Sensitivity = 3 / (3 + 0) = 1.0
Perfect sensitivity! The model captured ALL positive cases. Note: There is one false positive at position 4 (predicted=1, actual=0), but this doesn't affect sensitivity—it would affect precision instead.
actual_labels = [1, 1, 1, 0, 0]
predicted_labels = [0, 0, 0, 1, 1]0.0This represents the worst-case scenario:
• Total actual positives: 3 instances (positions 0, 1, 2) • True Positives: None of the actual positives were correctly identified → 0 instances • False Negatives: All 3 positive samples were missed → 3 instances
Sensitivity = 0 / (0 + 3) = 0.0
The model failed to identify ANY positive cases. Interestingly, the model also predicted all negatives as positives, suggesting the predictions might be inverted or the model is fundamentally flawed.
Constraints