Platt Scaling

Background

Post-hoc calibration via Platt scaling fits a 2-parameter logistic on top of raw model scores so the resulting probabilities match observed frequencies. It is the cheapest, most-cited fix for the canonical "AUC is great but the probabilities are wrong" failure mode: held-out scores in, calibrated probabilities out, no retrain.

Problem statement

Implement platt_fit(scores, labels, n_iter=500, lr=0.5) that fits parameters $(a, b)$ minimising binary cross-entropy:

$$\hat{p}_i \;=\; \sigma(a\,s_i + b), \qquad \mathcal{L}(a, b) \;=\; -\frac{1}{N}\sum_i \bigl[y_i \ln \hat{p}_i + (1 - y_i)\ln(1 - \hat{p}_i)\bigr]$$

via gradient descent. Return (a, b) as two Python floats. Initialise $(a, b) = (1, 0)$.

Input

• scores - array-like of floats, raw model scores (any real range).
• labels - array-like of $\{0, 1\}$, same length as scores.
• n_iter - int, gradient steps (default $500$).
• lr - float, learning rate (default $0.5$).

Output

• (a, b) - tuple of plain Python floats.

Examples

Example 1 - perfectly separable labels -> large |a|, b shifts to threshold

Input:  scores = [-3, -2, -1, 1, 2, 3], labels = [0, 0, 0, 1, 1, 1]
Output: (a, b) such that sigmoid(a*s + b) ~= labels

Constraints

• scores and labels must be the same length, length $\ge 2$.
• Use the numerically stable sigmoid for negative arguments: 1 - sigmoid(-z) for $z < 0$, or equivalently np.where(z >= 0, 1/(1+exp(-z)), exp(z)/(1+exp(z))). Tests use long-tailed scores that overflow naive exp.
• Tests check that after fitting, the mean of $\hat{p}$ matches the mean of $y$ to within $0.05$ and that the predictions correctly order the classes on the separable example.

Notes

• Why $(1, 0)$ init. Starting at the identity ($a = 1, b = 0$) means the first iteration's gradient is dominated by the score-label mismatch, not by an arbitrary scale.
• Newton vs gradient descent. Production Platt fits use a Newton step (the Hessian is 2x2 and trivially invertible) - convergence in 5-10 iterations vs hundreds for plain GD. The contract here uses GD for clarity; the variants section names Newton as the upgrade.