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2421Bayes Act Under Weighted Squared Loss 1Suppose the loss is L(a,Y)=W(Y-a) 2 where W>0 is observed at prediction time. What predictor minimizes E[L(a,Y)|X,W]?机器学习简单derivation未尝试免费2422Log-Loss Gap Between Two Positive ForecastsAn event occurs (y=1). Forecast A assigns probability 0.9 and forecast B assigns probability 0.7. By how much is B's log loss larger than A's?机器学习简单数值题未尝试免费2423Weighted Log-Loss Bayes Probability 3For binary Y with P(Y=1|X)=p, consider weighted log-loss L(q,Y) = -alpha Y ln q - beta (1-Y) ln(1-q). What probability q minimizes the conditional expected loss?机器学习中等derivation未尝试免费2424Convexity of the Log-Cosh Loss 4Show that ell(r)=ln cosh(r) is convex in the residual r.机器学习中等derivation未尝试免费2425Why Asymmetric Loss Changes the Optimal Prediction 21Why does changing the relative penalty on overprediction versus underprediction generally move the Bayes act away from the conditional mean?机器学习困难essay未尝试面试订阅2426Why Robust Losses Matter Under Contamination 22Why might a practitioner prefer Huber or pseudo-Huber loss when the data pipeline occasionally produces corrupted labels or sensor spikes?机器学习简单essay未尝试免费2427Decision Threshold Under Asymmetric Classification CostA false negative costs 5 and a false positive costs 1. If p is the predicted probability of the positive class, above what threshold should you classify as positive?机器学习中等derivation未尝试面试订阅2428Weighted-Absolute-Loss Constant ForecastDemand Y takes values 0, 2, and 5 with probabilities 0.5, 0.3, and 0.2. Under a loss of 3(Y-a) + + 1(a-Y) +, what constant forecast a minimizes expected loss?机器学习中等derivation未尝试面试订阅2429Total Huber Loss on a Residual SetUsing Huber loss with delta = 1, compute the total loss on residuals 0.5, -1.2, and 3.0.机器学习中等数值题未尝试面试订阅2430Why the Weighted-Brier Bayes Act Is Still a Weighted Mean 7For binary Y and weighted squared loss alpha Y (1-q) 2 + beta (1-Y) q 2, derive the Bayes probability q as a function of p=P(Y=1|X).机器学习困难derivation未尝试面试订阅2431Pseudo-Huber Gradient 8For pseudo-Huber loss ell(r)=delta 2(sqrt(1+(r/delta) 2)-1), derive d ell / d r.机器学习简单derivation未尝试免费2432Squared-Loss Bayes Act for a Discrete TargetY takes values 1, 1, 4, and 7 with equal probability. What constant forecast minimizes expected squared loss?机器学习简单derivation未尝试免费2433Pinball Loss Subgradient at the Kink 9For pinball loss rho tau(r)=tau r if r>=0 and (tau-1)r if r<0, what is the subgradient set at r=0?机器学习中等derivation未尝试面试订阅2434Why a Strictly Proper Loss Must Prefer Truthful Probabilities 10Why is it desirable for a probabilistic classification loss to be strictly proper rather than merely classification-accurate?机器学习困难derivation未尝试面试订阅2435Why Log-Loss Punishes Overconfidence More Than Brier LossWhy does log-loss react much more harshly than Brier loss when a model assigns near-certainty to the wrong class?机器学习困难essay未尝试面试订阅2436Why the Business Objective May Differ From the Training Loss 23Why can it be rational to train a model under one loss and evaluate the final decision under a different business metric?机器学习简单essay未尝试免费2437Why Huber Sits Between Squared and Absolute LossWhy is Huber loss often described as sitting 'between' squared loss and absolute loss?机器学习中等essay未尝试面试订阅2438Why Convexity Makes Averaging Predictions SafeWhy does convexity of a loss function support the intuition that averaging similar predictors often cannot hurt too much?机器学习困难essay未尝试面试订阅2439Why Asymmetric Loss Moves the Target Away From the MeanWhy does an asymmetric loss generally make the optimal constant prediction move away from the mean of the target distribution?机器学习困难essay未尝试面试订阅2440Why Heavy-Tailed Noise Pushes You Away From Pure Squared Loss 15Why is pure squared loss often a poor default when the residual distribution has rare but huge outliers?机器学习中等derivation未尝试面试订阅