Parametric methods¶

Parametric calibrators repair miscalibration with a map chosen from a small, named family — two or three interpretable parameters fitted by maximum likelihood. On the small calibration sets typical of credit risk they are usually the right first choice: few parameters mean low variance, the fitted values have direct diagnostic readings, and an auditor can reproduce the map from the reported parameters alone. This chapter derives the three families implemented in probcal.parametric — Platt scaling, temperature scaling, and beta calibration — and develops the interpretation of every parameter.

Throughout, \( s \in (0, 1) \) is the raw score being calibrated, \( z = \operatorname{logit}(s) = \ln\frac{s}{1-s} \) its logit, \( \sigma(t) = 1/(1 + e^{-t}) \) the logistic function, and \( g(s) \) the calibrated probability. Following probcal's input convention, calibrators accept probabilities and convert to logits internally; users holding raw logits apply the exported expit first.

Platt scaling¶

Platt (1999) faced scores that were not probabilities at all — support vector machine margins — and proposed passing them through a fitted sigmoid. In probcal's formulation the feature is the logit of the score, so the map is

\[ g(s) = \sigma\bigl(a z + b\bigr), \qquad z = \operatorname{logit}(s), \]

with slope \( a \) and intercept \( b \) fitted by maximum likelihood — an ordinary logistic regression of the outcome on a single covariate. PlattCalibrator fits it with the shared IRLS core (probcal._math.irls_logistic), which detects quasi-separation and falls back to a ridge-stabilized solution rather than diverging.

Fitting targets. Platt's original recipe does not regress on the hard labels \( y \in \{0, 1\} \). To keep the fitted sigmoid away from degenerate solutions when one class is scarce, the positive and negative targets are smoothed to

\[ t_{+} = \frac{N_{+} + 1}{N_{+} + 2}, \qquad t_{-} = \frac{1}{N_{-} + 2}, \]

where \( N_{+} \) and \( N_{-} \) count positive and negative calibration observations. The smoothing has a Bayesian reading — it is the posterior mean of a uniform-prior Bernoulli probability within each class — and its practical value on small samples, together with a numerically robust fitting procedure, is analyzed by Lin, Lin and Weng (2007), whose treatment probcal follows. With a 3% event rate and 500 calibration points, \( N_{+} \approx 15 \), and the difference between regressing on \( \{0,1\} \) and on the smoothed targets is material.

Interpretation. The slope corrects the spread of the score distribution: \( a < 1 \) shrinks over-dispersed, overconfident scores toward the base rate, while \( a > 1 \) sharpens underconfident ones. The intercept moves every prediction by \( b \) log-odds units — it is calibration-in-the-large repair, the same role the standalone offset plays. The identity map corresponds exactly to \( (a, b) = (1, 0) \), so departures of the fitted pair from \( (1, 0) \) are the diagnosis: probcal's interpret() states it in these terms, and the calibration slope and intercept metrics of probcal.metrics.regression are the same two quantities estimated as diagnostics rather than repairs.

A caveat on families. Fitted on logits, as here, the logistic family contains the identity \( (a, b) = (1, 0) \): if the input is already calibrated, maximum likelihood can leave it alone. Fitted on raw scores \( s \) directly — Platt's original setting, where \( g(s) = \sigma(As + B) \) — the family contains no identity on \( (0,1) \), so the calibrator distorts even perfectly calibrated inputs. This distinction, emphasized by Kull, Silva Filho and Flach (2017), is the design reason probcal fits on the logit scale and one of the two motivations for beta calibration below.

Temperature scaling¶

Temperature scaling (Guo et al., 2017) is Platt scaling with the intercept removed and the slope reparameterized:

\[ g(s) = \sigma\!\left(\frac{z}{T}\right), \qquad T > 0 . \]

The single parameter \( T \) is fitted by minimizing the negative log-likelihood on the calibration set. The objective is smooth and, on the reparameterization \( u = 1/T \), convex — TemperatureCalibrator solves it with a guarded one-dimensional Newton iteration (probcal._math.newton_1d) and falls back to bisection when the Newton step leaves the bracket.

Interpretation. \( T > 1 \) divides every logit by more than one, pulling probabilities toward \( 1/2 \): the model was overconfident and is being softened. \( T < 1 \) sharpens an underconfident model. \( T = 1 \) is the identity. Because there is no intercept, temperature scaling cannot move the base rate: the score \( s = 1/2 \) maps to \( 1/2 \) for every \( T \), and more generally the map is symmetric about the logit origin. A model whose only defect is a portfolio-level bias — the common credit-risk situation after a shift in central tendency — is beyond temperature's reach; use Platt scaling or the offset for that. Guo et al. (2017) found this one-parameter family remarkably effective for modern neural networks, whose miscalibration is often close to a pure confidence rescaling; on structured tabular scores with asymmetric distortion, expect the richer families below to win.

Degrees of freedom. Platt has two parameters, temperature one. Under the nested-validation selector ties on the scoring criterion break toward fewer parameters, so temperature is preferred exactly when the data cannot distinguish the two — the parsimony principle applied to calibration.

Beta calibration¶

Beta calibration (Kull, Silva Filho and Flach, 2017) starts from a generative question: if the score distributions within each class are Beta distributions — the natural family for quantities living on \( (0,1) \) — what does the true calibration map look like? Let

\[ s \mid Y = 1 \sim \mathrm{Beta}(\alpha_1, \beta_1), \qquad s \mid Y = 0 \sim \mathrm{Beta}(\alpha_0, \beta_0), \]

with class prior \( \pi = \Pr(Y = 1) \). Bayes' rule gives \( \operatorname{logit} \Pr(Y = 1 \mid s) = \ln \frac{f_1(s)}{f_0(s)} + \operatorname{logit}\pi \), and taking the log-ratio of the two Beta densities,

\[ \ln \frac{f_1(s)}{f_0(s)} = (\alpha_1 - \alpha_0) \ln s \;-\; (\beta_0 - \beta_1) \ln(1 - s) \;+\; \text{const}. \]

Writing \( a = \alpha_1 - \alpha_0 \), \( b = \beta_0 - \beta_1 \), and absorbing the constants and the prior into \( c \), the calibration map is

\[ \operatorname{logit} g(s) = a \ln s \;-\; b \ln(1 - s) \;+\; c , \]

a logistic regression on the two features \( \ln s \) and \( -\ln(1-s) \). This is the "abm" variant of BetaCalibrator — the full three-parameter family. The constrained variants tie or drop parameters: "ab" imposes \( a = b \), which collapses the two features into \( a \,\operatorname{logit}(s) + c \) — exactly Platt scaling on logits — and "a" retains a single free exponent (the exact tying is fixed, with a DECISIONS entry, when the calibrator is implemented). Temperature scaling is the further special case \( a = b = 1/T \), \( c = 0 \). The parametric families of this chapter thus form a nested hierarchy, which is what makes parsimony tie-breaking in the selector coherent.

The identity property. Setting \( (a, b, c) = (1, 1, 0) \) gives \( \operatorname{logit} g(s) = \ln s - \ln(1-s) = \operatorname{logit}(s) \): the identity map is an interior point of the beta family. Kull, Silva Filho and Flach (2017) stress the consequence: beta calibration cannot un-calibrate an already calibrated model beyond sampling noise, because maximum likelihood at the identity has no gradient pushing away from it. The same authors' journal-length treatment (Kull et al., Electronic Journal of Statistics, 2017) develops the full theory, and the multiclass generalization appears as Dirichlet calibration (Kull et al., 2019) — out of probcal's binary scope but the natural pointer for multiclass readers.

Monotonicity constraint. The map is non-decreasing in \( s \) iff \( a \ge 0 \) and \( b \ge 0 \). Unconstrained maximum likelihood can violate this on noisy or tiny calibration sets, producing a map that reverses ranking somewhere — unacceptable when downstream decisions assume order preservation. probcal enforces the constraint by the refit strategy of the reference betacal implementation: if the unconstrained fit yields \( a < 0 \), the \( \ln s \) feature is dropped (fixing \( a = 0 \)) and the model refitted; symmetrically for \( b < 0 \). The fitted object records which constraint was active, interpret() reports it, and the behavior is logged as a DECISIONS entry.

Interpretation. The exponents control tail sensitivity independently — the asymmetry that Platt and temperature cannot express. Near \( s \to 0 \) the map behaves like \( g(s) \propto s^{a} \), so \( a \) governs the low-probability tail: for a PD model, \( a < 1 \) flattens and raises the smallest PDs (the model was too eager to assign near-zero risk), \( a > 1 \) deepens them. Near \( s \to 1 \) the mirrored role belongs to \( b \). The intercept \( c \) shifts all predictions in log-odds, as in Platt. A fit with \( a \approx b \) says the distortion was symmetric and the extra parameter was not needed; \( a \neq b \) quantifies exactly the kind of one-sided tail distortion that low-event-rate portfolios exhibit.

Parameter interpretation at a glance¶

Calibrator	Parameters	Identity at	Reading of departures from identity
`PlattCalibrator`	\( a, b \)	\( (1, 0) \)	\( a \lessgtr 1 \): over/underconfident spread; \( b \neq 0 \): base-rate shift in log-odds
`TemperatureCalibrator`	\( T \)	\( T = 1 \)	\( T > 1 \): overconfident (soften); \( T < 1 \): underconfident (sharpen); base rate untouchable
`BetaCalibrator`	\( a, b, c \)	\( (1, 1, 0) \)	\( a \): low-\(s\) tail sensitivity; \( b \): high-\(s\) tail sensitivity; \( c \): base-rate shift; \( a \neq b \): asymmetric distortion

Every fitted calibrator returns this reading through interpret(), populated with the actual fitted values — the table is the contract for what those interpretations must say.

A worked reading¶

Numbers make the interpretation contract concrete. Suppose a PD model scores a calibration set with a realized default rate of 3.1%, and the fitted calibrators come back as follows.

BetaCalibrator(variant="abm") reports \( (a, b, c) = (0.82,\; 1.31,\; -0.42) \). Reading each parameter: \( a = 0.82 < 1 \) means that near the low-score tail the map behaves like \( s^{0.82} \), which decays more slowly than \( s \) itself — the smallest raw PDs are pulled up. The model was too confident about its safest accounts, a familiar pattern when the development sample under-represents rare defaults among high-quality obligors. \( b = 1.31 > 1 \) sharpens the approach to the high-risk end: raw scores near the top understated risk. \( c = -0.42 \) multiplies every odds by \( e^{-0.42} \approx 0.66 \), correcting a portfolio-level overestimation of about a third in odds terms. The asymmetry \( a \neq b \) is the substantive finding — the distortion is different in the two tails, so no symmetric family could have repaired it fully.

On the same data PlattCalibrator reports \( (a, b) = (1.05, -0.38) \). It captures nearly the same base-rate correction (the intercept), sees an almost-correct spread on average (\( a \approx 1 \)), and is structurally blind to the tail asymmetry: the single slope averages the too-flat low tail against the too-steep high tail. If out-of-fold log loss barely distinguishes the two fits, the asymmetry was not estimable from this sample and parsimony favors Platt; if beta wins clearly, the tails carried real signal. This is precisely the comparison the selector automates, and the reason its report shows the per-candidate scores rather than a bare winner.

TemperatureCalibrator on the same data is the wrong tool: whatever \( T \) it fits, it cannot express the \( -0.42 \) base-rate correction, and its fitted value will contort to compromise between the spread and the shift. A fitted \( T \) far from 1 accompanied by a failing calibration-in-the-large guardrail (see Metrics and tests) is the telltale of this misuse.

Choosing within the parametric family¶

The nesting temperature ⊂ Platt ⊂ beta orders the families by flexibility, and the choice is a bias–variance decision made honestly by out-of-fold log loss in the selector. Two rules of thumb survive contact with practice. First, if the diagnosed defect is purely a base-rate shift, none of these families is the parsimonious answer — a one-parameter logit offset fixes calibration-in-the-large without touching spread, and it composes transparently with any calibrator. Second, when the reliability diagram on the logit scale (see Visualization) shows curvature — not just a wrong slope — the distortion is outside every family in this chapter, and the nonparametric methods of the next chapter apply.

In probcal¶

from probcal import BetaCalibrator, PlattCalibrator, TemperatureCalibrator

platt = PlattCalibrator().fit(s_cal, y_cal)
print(platt.a_, platt.b_)                  # identity is (1, 0)

temp = TemperatureCalibrator().fit(s_cal, y_cal)
print(temp.T_)                             # identity is 1; cannot move the base rate

beta = BetaCalibrator(variant="abm").fit(s_cal, y_cal)
print(beta.interpret())                    # a, b, c against the identity (1, 1, 0)

p = beta.predict_proba(s_new)
print(platt.affine_logit_coeffs_)          # (a, b) — the affine-exact attribution hook

References¶

Guo, C., Pleiss, G., Sun, Y., Weinberger, K. Q. (2017). "On Calibration of Modern Neural Networks." ICML, PMLR 70, 1321–1330.
Kull, M., Silva Filho, T., Flach, P. (2017). "Beta calibration: a well-founded and easily implemented improvement on logistic calibration for binary classifiers." AISTATS, PMLR 54, 623–631.
Kull, M., Silva Filho, T., Flach, P. (2017). "Beyond sigmoids: How to obtain well-calibrated probabilities from binary classifiers with beta calibration." Electronic Journal of Statistics 11(2), 5052–5080.
Kull, M., Perello-Nieto, M., Kängsepp, M., Silva Filho, T., Song, H., Flach, P. (2019). "Beyond temperature scaling: Obtaining well-calibrated multi-class probabilities with Dirichlet calibration." NeurIPS 32.
Lin, H.-T., Lin, C.-J., Weng, R. C. (2007). "A Note on Platt's Probabilistic Outputs for Support Vector Machines." Machine Learning 68(3), 267–276.
Platt, J. C. (1999). "Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods." In Advances in Large Margin Classifiers, MIT Press, 61–74.