# Reason About an Intercept Shift in Logistic Regression A trained binary classifier produces `p(x) = sigmoid(w^T x + b)` and predicts the positive class when `p(x) >= t`, for a fixed threshold `0 < t < 1`. Without retraining `w`, the intercept is changed from `b` to `b + 10`. Explain what changes and what does not. In particular, reason carefully about predicted-positive count, recall, false-positive rate, precision, ranking metrics, and calibration. Do not assume a balanced data set or claim a direction for a metric unless it follows from the stated conditions. ### Constraints & Assumptions - Evaluation is performed on the same fixed labeled examples before and after the change. - The classification threshold `t` is held fixed. - Scores and labels are not otherwise modified. ### Clarifying Questions to Ask - Is the threshold fixed in probability space or retuned after the shift? - Are we discussing sample metrics on a fixed set or expected population metrics? - Is the model score used only for classification, or also as a calibrated probability? ### What a Strong Answer Covers - The equivalent shift in log-odds and decision boundary - Monotonic changes that can be proven for a fixed sample - Why precision has no guaranteed direction - Ranking invariance versus calibration damage - How a threshold adjustment could restore the original decisions ### Follow-up Questions - What threshold in logit space reproduces the original classifications? - Does ROC AUC change after adding a constant to every logit? - What happens when nearly every example becomes positive? - How would you diagnose whether an intercept correction is actually warranted?