First, clarify the confusion matrix counts at each threshold (TP, FP, FN, TN) and confirm the total of 10,000 examples. Then compute precision, recall, and F1 using their standard formulas, and compare the results to discuss the trade-off between precision and recall when lowering the threshold.
Pro tip: Always state the formulas explicitly and interpret the metrics in the context of the 1% prevalence—this shows you understand that high accuracy can be misleading and that precision/recall are more informative for imbalanced data.
From the provided confusion matrix at each threshold, identify TP, FP, FN, TN. Verify that the sum equals 10,000 and that the number of actual positives (TP+FN) is 100 (1% of 10,000).
Calculate precision = TP / (TP + FP) and recall = TP / (TP + FN) for each threshold. Show the calculations clearly.
Calculate F1 = 2 * (precision * recall) / (precision + recall) for each threshold. If precision and recall are both zero, F1 is zero.
Compare the metrics across thresholds (0.50 vs 0.20). Discuss how lowering the threshold typically increases recall but decreases precision, and what that means for the business context (e.g., TikTok content moderation).
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At 0.50: cost = 40*1 + 40*20 = 40 + 800 = 840.
First, clarify the threshold range and the underlying score distributions (or confusion matrices) at each threshold. Then compute expected cost as FP*1 + FN*20 for each threshold, and identify the minimum. If distributions are not given, state assumptions and demonstrate the calculation with a hypothetical example.
Pro tip: Emphasize that the optimal threshold depends on the cost ratio (20:1) and the model's ROC curve; a common mistake is to default to 0.5, but here you should shift the threshold lower to reduce costly false negatives.
Ask for or state the score distributions for positive and negative classes, or the confusion matrix at each threshold. If not provided, assume a simple example (e.g., uniform distributions) to illustrate the method.
Expected cost = (Number of false positives) * 1 + (Number of false negatives) * 20. Alternatively, use rates: cost = FP_rate * P(negative) * 1 + FN_rate * P(positive) * 20.
For each threshold, determine FP and FN counts (or rates) from the distributions, then plug into the formula. Create a table of threshold vs. cost.
Compare the costs across thresholds and select the one with the lowest expected total cost. Explain why it makes sense given the 20:1 cost asymmetry.
Mention that the optimal threshold minimizes expected cost, not accuracy. Discuss how changing the cost ratio would shift the threshold, and note any practical constraints (e.g., business rules).
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My answer here was decent conceptually but I struggled to make the numeric intuition crisp.
Start by defining ROC-AUC and PR-AUC, then explain how class imbalance affects each metric differently. Use a concrete confusion matrix example with numbers to illustrate why ROC-AUC can remain high while PR-AUC drops, showing that PR-AUC better reflects performance on the minority class. Conclude with when to use each metric and the implications for model selection.
Pro tip: Emphasize that in imbalanced settings, the baseline for PR-AUC is the prevalence of the positive class, so a PR-AUC of 0.1 might be meaningful if prevalence is 0.01, whereas ROC-AUC's baseline is always 0.5. This nuance shows deep understanding and avoids misinterpreting PR-AUC values.
Briefly define ROC-AUC as the area under the ROC curve (TPR vs. FPR) and PR-AUC as the area under the precision-recall curve. Mention that ROC-AUC is threshold-independent and measures overall separability, while PR-AUC focuses on the positive class.
Explain that with heavy imbalance, the large number of true negatives can inflate the FPR denominator, making ROC-AUC look good even if the model performs poorly on the positive class. PR-AUC, however, ignores true negatives and directly measures precision and recall for the positive class.
Use a concrete example: suppose 1000 samples, 990 negatives, 10 positives. A model that predicts all negatives has TPR=0, FPR=0, but ROC-AUC is undefined (or 0.5 if we consider random). Better: a model with 5 TP, 5 FN, 50 FP, 940 TN. Compute TPR=0.5, FPR=0.05, so ROC point is (0.05,0.5). ROC-AUC might be high (e.g., 0.9) if other thresholds perform well, but precision=5/55≈0.09, recall=0.5, so PR-AUC is low. Show that high ROC-AUC doesn't guarantee good precision.
Highlight that PR-AUC is more sensitive to improvements in the minority class because it doesn't get 'diluted' by true negatives. ROC-AUC can be overly optimistic when negatives dominate. Also note that PR-AUC's baseline is the positive class prevalence, so it's a more informative baseline.
State that for imbalanced datasets, PR-AUC is preferred when the positive class is the focus (e.g., fraud detection, click-through rate prediction). ROC-AUC can still be useful for comparing models if the class distribution is not extreme, but PR-AUC gives a clearer picture of performance on the minority class.
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Start by defining the expected cost as a function of the threshold, using the calibrated probabilities and the class prior. Then minimize this expected cost by taking the derivative and solving for the threshold, which yields a simple ratio of costs. Finally, express the threshold in terms of the given costs and prior, and discuss any assumptions or practical considerations.
Pro tip: Emphasize that the optimal threshold depends only on the cost ratio and the class prior, not on the model's calibration beyond being perfectly calibrated. This shows you understand the decision-theoretic foundation and can separate model quality from decision rules.
Write the expected cost of predicting positive vs. negative for a given instance, using the calibrated probability p(y=1|x) and the costs C_FP and C_FN. Incorporate the class prior if needed to express the probability in terms of the model output.
The optimal decision is to predict positive if the expected cost of predicting positive is less than or equal to that of predicting negative. This yields an inequality involving the probability and the costs.
Rearrange the inequality to isolate the probability p(y=1|x). The threshold t* is the value of p at which the two expected costs are equal.
If the model outputs are calibrated probabilities, then t* is directly the probability threshold. If the model outputs a score that is a monotonic function of the probability, adjust accordingly. Use the class prior to relate the probability to the likelihood ratio if necessary.
Present the final expression: t* = C_FP / (C_FP + C_FN) when the model is perfectly calibrated and the prior is already incorporated in the calibration. If the prior is separate, the threshold on the likelihood ratio or on the calibrated probability may include the prior. Clarify the assumptions.
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