Multiclass Calibration

library(probcal)

From two classes to several

The binary calibrators in probcal take a vector of scores or probabilities. The multiclass calibrators take a matrix with one row per observation and one column per class. The same functions serve both settings: a vector input is treated as binary, and a matrix input is treated as multiclass. Labels are a factor or a vector of integer class codes in 1:K, where K is the number of columns.

A multiclass model usually returns either a matrix of logits or a matrix of softmax probabilities. Temperature scaling and vector scaling work on logits. Dirichlet calibration works on probability matrices. The one-vs-rest wrapper uses the input scale required by its binary base method: scores or probabilities for Platt scaling, probabilities for beta calibration, isotonic regression, and histogram binning, and logits for temperature scaling.

Simulating an overconfident classifier

We simulate true class probabilities for three classes, draw labels from them, and then sharpen the probabilities to mimic an overconfident model. The calibration split fits the calibrator and the test split evaluates it.

set.seed(2024)
n <- 1200
k <- 3

true_prob <- matrix(stats::runif(n * k), ncol = k)
true_prob <- true_prob / rowSums(true_prob)
labels <- apply(true_prob, 1, function(row) sample.int(k, 1, prob = row))

# An overconfident model: push probabilities toward 0 and 1.
sharpen <- function(p, power = 2.5) {
  q <- p^power
  q / rowSums(q)
}
raw_prob <- sharpen(true_prob)
raw_logits <- log(pmax(raw_prob, 1e-12))

split <- sample(rep(c("calibration", "test"), each = n / 2))
cal <- split == "calibration"
test <- split == "test"

Measuring multiclass calibration

The calibration metrics accept the probability matrix and a type argument. The classwise form averages the binary calibration error over the one-vs-rest columns. The confidence form looks only at the top-label probability and whether the predicted class is correct.

ece(raw_prob[test, ], labels[test], type = "classwise")
#> [1] 0.121496
ece(raw_prob[test, ], labels[test], type = "confidence")
#> [1] 0.1676213
mmce(raw_prob[test, ], labels[test])
#> [1] 0.1194958

Temperature scaling on logits

Temperature scaling estimates a single positive scalar. Dividing every logit by the same value does not change the predicted class, so temperature scaling only sharpens or softens the probabilities.

temp_fit <- cal_temperature(raw_logits[cal, ], labels[cal])
temp_fit
#> 
#> ── ⚖ probcal calibrator ────────────────────────────────────────────────────────
#> Method: multiclass temperature scaling
#> Observations: 600
#> Input: logits (matrix)
#> Classes: 3

temp_pred <- predict(temp_fit, raw_logits[test, ])
ece(temp_pred, labels[test], type = "classwise")
#> [1] 0.03874575

Dirichlet calibration on probabilities

Dirichlet calibration is the multiclass generalization of beta calibration. It fits a linear map on the log-probabilities, regularized by an off-diagonal and intercept penalty whose strength is chosen by cross-validation when lambda is left at its default.

dir_fit <- cal_dirichlet(raw_prob[cal, ], labels[cal])
dir_pred <- predict(dir_fit, raw_prob[test, ])
ece(dir_pred, labels[test], type = "classwise")
#> [1] 0.03913495

One-vs-rest calibration

The one-vs-rest wrapper lifts any binary calibrator to several classes. It fits a binary calibrator that separates each class from the rest, applies them column by column, and renormalizes each row to sum to one.

ovr_fit <- cal_ovr(raw_prob[cal, ], labels[cal], method = "isotonic")
ovr_pred <- predict(ovr_fit, raw_prob[test, ])
ece(ovr_pred, labels[test], type = "classwise")
#> [1] 0.04916152

Comparing the calibrators

data.frame(
  method = c("raw", "temperature", "dirichlet", "one-vs-rest"),
  classwise_ece = c(
    ece(raw_prob[test, ], labels[test], type = "classwise"),
    ece(temp_pred, labels[test], type = "classwise"),
    ece(dir_pred, labels[test], type = "classwise"),
    ece(ovr_pred, labels[test], type = "classwise")
  )
)
#>        method classwise_ece
#> 1         raw    0.12149600
#> 2 temperature    0.03874575
#> 3   dirichlet    0.03913495
#> 4 one-vs-rest    0.04916152

Reliability diagram

For a probability matrix the reliability diagram draws one panel per class in the classwise layout, or a single panel of top-label confidence in the confidence layout.

reliability_diagram(dir_pred, labels[test], bins = 10, type = "classwise")

Out-of-fold calibration

When calibration data are scarce, cal_cv() fits the calibrator with out-of-fold predictions. It accepts a matrix input and the multiclass methods "temperature", "vector", "dirichlet", and "ovr".

cv_fit <- cal_cv(
  raw_prob,
  labels,
  method = "dirichlet",
  folds = 5,
  seed = 1
)
ece(cv_fit$oof_predictions, labels, type = "classwise")
#> [1] 0.0253298

Scope

The multiclass methods cover temperature scaling, vector scaling, Dirichlet calibration, and one-vs-rest calibration. Bayesian binning, near-isotonic ensembles, object detection calibration, and regression uncertainty calibration are future work.