Calibration Is Not Classification Accuracy

"The model is 94% accurate." It is the number that gets quoted, and on its own it tells you almost nothing about whether the model's confidence can be used for anything.

Accuracy and calibration are different properties. A model can have either without the other, and systems that route work based on confidence depend on the second one, not the first.

They come apart in both directions

Accurate, badly calibrated. Modern deep networks are systematically overconfident — Guo et al. showed that as architectures got deeper and more accurate, calibration got worse. A network that is right 94% of the time while reporting 0.99 on nearly everything is accurate and its confidence is worthless for triage.

Well calibrated, useless. A model that predicts the base rate for every input is perfectly calibrated. If 12% of cases are positive and it always says 0.12, its stated probabilities are exactly right and it has learned nothing. Calibration alone is not a quality measure.

Both properties have to be reported. Either one alone can be made to look good while the system fails.

Measuring calibration

Bin predictions by confidence, then compare the mean confidence in each bin against the observed accuracy in that bin.

import numpy as np


def reliability(probs, labels, bins=10):
    edges = np.linspace(0.0, 1.0, bins + 1)
    rows = []
    for lo, hi in zip(edges[:-1], edges[1:]):
        mask = (probs > lo) & (probs <= hi)
        if not mask.any():
            continue
        rows.append({
            "range": (lo, hi),
            "n": int(mask.sum()),
            "confidence": float(probs[mask].mean()),
            "observed": float(labels[mask].mean()),
        })
    return rows


def expected_calibration_error(rows, n_total):
    return sum(r["n"] / n_total * abs(r["confidence"] - r["observed"]) for r in rows)

Two cautions about expected calibration error, because it is the number everyone reports:

• It is bin-count dependent. Fewer bins gives a smaller number. Report the bin count or the comparison is meaningless.
• It averages away the region you care about. Most predictions usually sit at the confident end, so ECE is dominated by bins where little is at stake. Always look at the reliability table itself, not only its summary.

Fixing calibration without retraining

Calibration is usually repairable after the fact, on held-out data, without touching the model.

Temperature scaling divides the logits by a single learned scalar $T$:

$$\hat{p}_i = \operatorname{softmax}\!\left(\frac{z_i}{T}\right)$$

One parameter, fitted on a validation set by minimising negative log-likelihood. It cannot change the ranking of predictions, so accuracy is exactly unchanged — which is what makes it safe.

Isotonic regression fits a monotonic mapping from score to probability. More flexible, needs more data, and can overfit a small validation set.

The order matters: fit calibration on a set the model never saw, and evaluate calibration on a third set. Fitting and reporting on the same data gives you a number that describes your fit rather than your model.

Thresholds are a decision

For multi-label problems, each class gets its own operating point, and choosing it is a policy question wearing a statistical costume.

Youden's J statistic picks the point maximising sensitivity plus specificity minus one:

$$J = \operatorname{sensitivity} + \operatorname{specificity} - 1$$

def youden_threshold(scores, labels, grid=np.linspace(0.01, 0.99, 99)):
    best, best_j = 0.5, -1.0
    for t in grid:
        pred = scores >= t
        tpr = (pred & (labels == 1)).sum() / max((labels == 1).sum(), 1)
        tnr = (~pred & (labels == 0)).sum() / max((labels == 0).sum(), 1)
        if (j := tpr + tnr - 1) > best_j:
            best, best_j = float(t), j
    return best, best_j

Youden weights a false positive and a false negative equally. That is almost never the real cost structure. On a screening task a missed finding costs far more than a false alarm, and the right threshold is lower than Youden's.

So the practice I follow is: compute Youden's J as a starting point, record it as the provenance of the threshold, and adjust deliberately against the actual cost asymmetry. Thresholds in a shared library should carry a comment saying what set they were fitted on and by what rule, because otherwise they become magic constants nobody dares change.

Three consequences that surprise people:

• Per class, not global. Optimal thresholds differ by class by large margins, driven by base rate and separability. A single global threshold is leaving performance on the table for every class but one.
• They expire. A threshold fitted on one population is a property of that population. Prevalence shifts move it.
• They are not calibration. Moving a threshold changes which predictions are positive. It does nothing to whether the reported probabilities mean what they say.

Multi-label needs sigmoid, not softmax

If several labels can be true at once, the output layer is a per-class sigmoid, and each class is its own binary problem with its own threshold and its own calibration curve.

Softmax forces the scores to sum to one, which encodes "exactly one of these is true". Applied to a multi-label problem it makes a second finding necessarily reduce the confidence in the first, which is not what the data says.

What to report

For any classifier whose confidence feeds a decision:

1. Accuracy, or the task-appropriate equivalent, on held-out data.
2. A reliability table, with the bin count stated.
3. ECE as a summary, never as the whole picture.
4. Per-class thresholds with the rule and the fitting set that produced them.
5. Whether calibration was fitted, on what, and evaluated on what.
6. The subgroups checked, because aggregate calibration hides per-group failure.

Six lines. A model card with these is more useful than one with a single headline number, and it is the difference between a confidence score that can route work and one that can only decorate an interface.

Related

Selective prediction is what a calibrated confidence is for: deciding which predictions to accept, review or defer. What a saliency map does not prove covers the other artefact that gets over-read. The transport uncertainty research treats calibration and ranking as separate properties and reports both.