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Expected calibration error summarizes how reported confidence differs from observed correctness. For top-label statistical classification, the population quantity compares correctness probability conditional on confidence with that confidence. A commonly reported finite-sample estimate groups predictions into confidence intervals and compares the mean confidence and accuracy in each interval.^[ref]

For a nonempty bin $B_m$ among $n$ predictions, defineHere ${\hat{p}}_i$ is confidence in the selected class, not its numerical label. The empirical statistic isEmpty bins have zero weight and need no undefined bin average. The cited paper uses equal-width bins.^[ref]

A synthetic example shows why the partition matters. Four predictions have confidences $0.55,0.65,0.85,0.95$ and correctness indicators $1,0,1,1$. With one bin, accuracy and mean confidence are both $0.75$, so the estimate is zero. With four equal-width bins on $[0,1]$, only the last two are populated. Their accuracy/confidence pairs are $(0.5,0.6)$ and $(1,0.9)$, each with half the observations. ThusThis is ten percentage points, despite unchanged predictions and labels.

A zero coarse-bin result can conceal opposing gaps; it does not establish perfect probability calibration or agreement within every subgroup. Report bin boundaries, sample size, confidence convention and scale. Reliability diagrams reveal where the gaps occur, while a scalar expected calibration error supplies their sample-weighted summary. It is distinct from the fraction of incorrect classifications.

Temperature scaling is a method of probability calibration that divides a model's neural network logits by one fitted positive scalar before applying the softmax function. The temperature is selected after model training, using labels in a held-out validation set to minimize the negative log-likelihood. It adjusts reported probabilities without retraining all model weights.^[ref]

For class scores $z_1,\dots ,z_K$ and $T>0$,For the synthetic two-class scores $(2,0)$, the first-class probability isThe selected class remains the first while its reported confidence decreases.

The preservation of the selected class follows directly: dividing by positive $T$ preserves every score ordering and equality, and exponentiation is strictly increasing. Thus the set of classes attaining the largest score is unchanged. This concerns ordering between classes within one example; it does not assert preservation of confidence ordering between different examples.

A motivating measurement in the original study compares reliability diagrams for a 110-layer image-classification network. Its reported expected calibration error changes from $12.67\%$ before calibration to $0.96\%$ after temperature scaling. This is a result for that experimental setup and binned metric, not a guarantee for every classifier or input distribution.^[ref]

Use separate data for fitting the temperature and for final evaluation. One scalar can improve an overall confidence mismatch but cannot represent every class-specific or input-dependent error. The name refers to this probability transformation; no physical temperature is being measured.