  The focal loss has demonstrated its effectiveness in many real-world
applications such as object detection and image classification, but its
theoretical understanding has been limited so far. In this paper, we first
prove that the focal loss is classification-calibrated, i.e., its minimizer
surely yields the Bayes-optimal classifier and thus the use of the focal loss
in classification can be theoretically justified. However, we also prove a
negative fact that the focal loss is not strictly proper, i.e., the confidence
score of the classifier obtained by focal loss minimization does not match the
true class-posterior probability and thus it is not reliable as a
class-posterior probability estimator. To mitigate this problem, we next prove
that a particular closed-form transformation of the confidence score allows us
to recover the true class-posterior probability. Through experiments on
benchmark datasets, we demonstrate that our proposed transformation
significantly improves the accuracy of class-posterior probability estimation.
