  The Wasserstein probability metric has received much attention from the
machine learning community. Unlike the Kullback-Leibler divergence, which
strictly measures change in probability, the Wasserstein metric reflects the
underlying geometry between outcomes. The value of being sensitive to this
geometry has been demonstrated, among others, in ordinal regression and
generative modelling. In this paper we describe three natural properties of
probability divergences that reflect requirements from machine learning: sum
invariance, scale sensitivity, and unbiased sample gradients. The Wasserstein
metric possesses the first two properties but, unlike the Kullback-Leibler
divergence, does not possess the third. We provide empirical evidence
suggesting that this is a serious issue in practice. Leveraging insights from
probabilistic forecasting we propose an alternative to the Wasserstein metric,
the Cram\'er distance. We show that the Cram\'er distance possesses all three
desired properties, combining the best of the Wasserstein and Kullback-Leibler
divergences. To illustrate the relevance of the Cram\'er distance in practice
we design a new algorithm, the Cram\'er Generative Adversarial Network (GAN),
and show that it performs significantly better than the related Wasserstein
GAN.
