  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 only directional
derivatives. Certain entropies may be further extended continuously to open
cones in normed spaces containing signed densities. The extended densities are
G\^ateaux 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, Hyv\"arinen entropy, and quadratic entropy.
