Derive and regularize logistic regression

You are building a churn propensity model with logistic regression. Answer precisely: 1) Starting from a Bernoulli likelihood, derive the logistic regression log‑likelihood and its gradient w.r.t. β. Show how L2 and L1 regularization modify the objective (MLE → MAP); write the new objective and gradients/subgradients. 2) List OLS assumptions for linear regression. For each that is relevant to GLMs (e.g., multicollinearity, omitted variables, measurement error, non‑IID), explain how violations manifest in logistic regression and how regularization/feature engineering or robust inference address them. 3) Your positives are 3% of samples. Compare class‑weighting vs. focal loss vs. threshold‑moving. How does each affect calibration? Describe a calibration check and a recalibration method (Platt vs. isotonic) and when you’d prefer each. 4) Define a temporal validation scheme that avoids leakage. Include: feature freeze date, out‑of‑time test window, and k‑fold strategy compatible with time. Specify the exact splits on a 6‑month dataset. 5) With correlated features, contrast L1 vs. L2 on sparsity, stability, and interpretability. Propose a workflow that yields a sparse, stable model with confidence intervals for odds ratios. 6) Give one business‑aligned decision rule for choosing the score threshold using asymmetric costs, and show how to compute the expected value uplift over a “message all” policy.