Explain MSE vs MAE, AUC, and imbalance handling

Answer every part briefly and exactly. 1) In a regression objective, when would you choose MAE instead of MSE? Contrast outlier robustness, gradient behavior near zero, and optimization effects; provide a concrete case where MSE does poorly but MAE is acceptable. 2) For a binary classifier with 1% positive prevalence, explain what ROC-AUC = 0.90 and PR-AUC = 0.25 mean. Which metric is more informative in this setting, and why? Explain how ROC can appear strong while PR stays weak; mention the score distributions that produce this. 3) For a neural network binary classifier, select an output activation and a loss. Explain how you would address class imbalance with class weights or focal loss, and describe how each alters the gradient contributions of positive versus negative examples. 4) If the business requires precision ≥ 0.50, describe exactly how you would choose a probability threshold on validation data and prevent optimistic bias (for example, nested CV or a hold-out).

Overview: This question tests a candidate's grasp of regression loss trade-offs (MAE vs MSE), classification metrics under severe class imbalance (ROC-AUC vs PR-AUC), neural network output activation and loss dynamics, and practical approaches for imbalance mitigation and probability-threshold selection in binary classification.