Existence and uniqueness of proper scoring rules

Abstract

To discuss the existence and uniqueness of proper scoring rules one needs to extend the associated entropy functions as sublinear functions to the conic hull of the prediction set. In some natural function spaces, such as the Lebesgue Lp-spaces over Rd, the positive cones have empty interior. Entropy functions defined on such cones have directional derivatives only, which typically exist on large subspaces and behave similarly to gradients. Certain entropies may be further extended continuously to open cones in normed spaces containing signed densities. The extended densities are Gâteaux differentiable except on a negligible set and have everywhere continuous subgradients due to the supporting hyperplane theorem. We introduce the necessary framework from analysis and algebra that allows us to give an affirmative answer to the titular question of the paper. As a result of this, we give a formal sense in which entropy functions have uniquely associated proper scoring rules. We illustrate our framework by studying the derivatives and subgradients of the following three prototypical entropies: Shannon entropy, Hyvarinen entropy, and quadratic entropy.