AI summaryⓘ
The authors study a method originally developed by Ishida et al. to estimate the lowest possible error of a binary classifier using probability-based labels. They build on work by Ushio et al., who found this estimator can be unreliable if the probabilities are not perfectly accurate and suggested a fix called isotonic calibration. The authors analyze how a common calibration method, temperature scaling, changes the estimator’s output, showing mathematically that adjusting temperature can produce almost any estimate without changing the classifier's actual error. Their findings explain why calibration is important for meaningful error estimates and show that the estimated error depends heavily on how probabilities are produced, not just on the classifier itself.
Bayes errorsoft-labelprobability calibrationtemperature scalingclassifier marginGaussian logit modelexpected calibration errorbinary classification
Abstract
The soft-label Bayes-error estimator beta(z) = E[min(z, 1-z)] of Ishida et al. estimates the irreducible error of a binary task directly from probability-valued labels. Recent work by Ushio et al. showed that this estimator is fragile when the probabilities are not the true posterior: even perfectly calibrated soft labels can yield a substantially inaccurate estimate, and they propose isotonic calibration as a consistent remedy. We complement that line of work by characterizing exactly how the most widely used post-hoc calibration map -- temperature scaling -- distorts the proxy. We prove an exact, model-free identity reducing the temperature-scaled proxy to the classifier's margin distribution, from which we obtain (i) strict monotonicity in the temperature and (ii) a continuous bijection from the temperature axis onto the open interval (0, 1/2), so that a fixed classifier -- with fixed decisions and fixed 0-1 error -- can be made to report any proxy value whatsoever. Under a Gaussian model of the logits we further derive a two-parameter closed form for the entire proxy-versus-temperature curve. Across CIFAR-10, Fashion-MNIST, and SVHN (eight binary tasks), the proxy varies by 56x to 980x at constant test error, the closed form reproduces the empirical curve to within 0.018, and the calibration temperature that minimizes the expected calibration error does not coincide with any stable proxy value. Our results give a precise, predictive account of the distortion whose existence motivates calibration-based remedies, and they reinforce the practical recommendation that a proxy value is meaningful only together with the mechanism that produced its probabilities.