 Expected calibration error

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.