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2521Intercept-Only Logistic MLE 1For an intercept-only logistic model with n 1 positives and n 0 negatives, what fitted probability p hat maximizes the log-likelihood?机器学习简单derivation未尝试免费2522Intercept From the Positive Rate 2In an intercept-only logistic model, if the fitted probability is p hat, what intercept b solves sigma(b)=p hat?机器学习简单derivation未尝试免费2523Gradient of Logistic Negative Log-Likelihood 3For one observation (x,y) with y in 0,1 and score z = w T x, what is the gradient of the negative log-likelihood with respect to w?机器学习中等derivation未尝试免费2524Why No Closed Form in Logistic Regression 5Why does logistic regression usually require iterative optimization rather than a normal-equation-style closed form?机器学习中等essay未尝试免费2525One Newton Step for an Intercept-Only Logistic ModelAn intercept-only logistic model is fit to 7 positives and 3 negatives. Starting from b 0 = 0, what is one Newton step b 1 for minimizing the negative log-likelihood?机器学习困难数值题未尝试面试订阅2526Why Separable Data Pushes Coefficients Outward 7Why do logistic-regression coefficients tend to diverge on perfectly linearly separable data if no regularization is used?机器学习简单essay未尝试免费2527Probability From a Logit Score 8If a logistic model outputs score z = ln 3, what probability does it assign to class 1?机器学习中等数值题未尝试免费2528Why Log-Loss Rewards Calibration 9Why does a well-calibrated probability forecaster typically fare better under log-loss than a forecaster that only gets rankings right?机器学习中等essay未尝试免费2529Why Regularization Helps Even When Logistic Is Convex 11If logistic loss is already convex, why can regularization still be crucial in practice?机器学习中等essay未尝试面试订阅2531Why One-vs-Rest Scores Need Not Sum to One 15Why can independently trained one-vs-rest logistic classifiers produce class probabilities that do not sum to one?机器学习简单essay未尝试免费2532Hessian of Logistic Negative Log-Likelihood 4For one observation with score z = w T x, what is the Hessian of the negative log-likelihood with respect to w?机器学习简单derivation未尝试免费2534One Gradient Step on a Tiny Logistic ProblemA one-feature logistic model without intercept uses beta = 0 initially, learning rate 0.2, data x = [-1, 0, 1], and labels y = [0, 0, 1]. What is beta after one gradient step on the negative log-likelihood?机器学习困难数值题未尝试面试订阅2535Decision Threshold Under Asymmetric Classification CostsA desk incurs cost 1 for a false positive and cost 5 for a false negative. Under a calibrated logistic probability p = P(Y=1|x), above what threshold should it predict class 1 to minimize expected cost?机器学习困难derivation未尝试面试订阅2536Why Softmax Fixes Joint Normalization 16What does the softmax construction add that one-vs-rest logistic models do not provide automatically?机器学习简单essay未尝试免费2537Why Logistic Probabilities Are Useful Downstream 18Why is it valuable that logistic regression produces a calibrated probability estimate rather than only a hard class label?机器学习中等essay未尝试面试订阅2538Why Logistic Beats Hard Threshold Rules for Training 23Why is a smooth probabilistic loss easier to optimize than training directly against a hard classification rule?机器学习中等essay未尝试免费2540Intercept Shift for a Deployment Prior ChangeA logistic model was trained under class prior 0.5 and has intercept -0.4. At deployment the base rate falls to 0.2 while feature likelihood ratios are assumed unchanged. What adjusted intercept should be used?机器学习困难数值题未尝试面试订阅2541One Gradient Step on a Single Logistic Observation 22For one observation with x = 2, y = 1, current weight w = 0, and learning rate eta = 0.4, what is one gradient-descent update on the negative log-likelihood?机器学习简单数值题未尝试免费