Calibration tests in multi-class classification: A unifying framework

In safety-critical applications a probabilistic model is usually required to be calibrated, i.e., to capture the uncertainty of its predictions accurately. In multi-class classification, calibration of the most confident predictions only is often not sufficient. We propose and study calibration measures for multi-class classification that generalize existing measures such as the expected calibration error, the maximum calibration error, and the maximum mean calibration error. We propose and evaluate empirically different consistent and unbiased estimators for a specific class of measures based on matrix-valued kernels. Importantly, these estimators can be interpreted as test statistics associated with well-defined bounds and approximations of the p-value under the null hypothesis that the model is calibrated, significantly improving the interpretability of calibration measures, which otherwise lack any meaningful unit or scale.