## Existence and Uniqueness of Proper Scoring Rules

*Evgeni Y. Ovcharov*; 16(Nov):2207−2230, 2015.

### 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 $L^p$-spaces
over $\mathbb{R}^d$, 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 Gateaux
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.

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