  The prediction accuracy of machine learning methods is steadily increasing,
but the calibration of their uncertainty predictions poses a significant
challenge. Numerous works focus on obtaining well-calibrated predictive models,
but less is known about reliably assessing model calibration. This limits our
ability to know when algorithms for improving calibration have a real effect,
and when their improvements are merely artifacts due to random noise in finite
datasets. In this work, we consider detecting mis-calibration of predictive
models using a finite validation dataset as a hypothesis testing problem. The
null hypothesis is that the predictive model is calibrated, while the
alternative hypothesis is that the deviation from calibration is sufficiently
large.
  We find that detecting mis-calibration is only possible when the conditional
probabilities of the classes are sufficiently smooth functions of the
predictions. When the conditional class probabilities are H\"older continuous,
we propose T-Cal, a minimax optimal test for calibration based on a debiased
plug-in estimator of the $\ell_2$-Expected Calibration Error (ECE). We further
propose Adaptive T-Cal, a version that is adaptive to unknown smoothness. We
verify our theoretical findings with a broad range of experiments, including
with several popular deep neural net architectures and several standard
post-hoc calibration methods. T-Cal is a practical general-purpose tool, which
-- combined with classical tests for discrete-valued predictors -- can be used
to test the calibration of virtually any probabilistic classification method.
