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Evgeni Y. Ovcharov

Publications and source records attributed to Evgeni Y. Ovcharov.

3 recordsLinked to original sources
arXiv2015

Proper Scoring Rules and Bregman Divergences

We revisit the mathematical foundations of proper scoring rules (PSRs) and Bregman divergences and present their characteristic properties in a unified theoretical framework. In many situations it is preferable not to generate a PSR directly from its convex entropy on the unit simplex but instead by the sublinear extension of the entropy to the positive orthant. This gives the scoring rule simply as a subgradient of the extended entropy, allowing for a more elegant theory. The other convex extensions of the entropy generate affine extensions of the scoring rule and induce the class of functional Bregman divergences. We discuss the geometric nature of the relationship between PSRs and Bregman divergences and extend and unify existing partial results. We also approach the topic of differentiability of entropy functions. Not all entropies of interest possess functional derivatives, but they do all have directional derivatives in almost every direction. Relying on the notion of quasi-interior of a convex set to quantify the latter property, we formalise under what conditions a PSR may be considered to be uniquely determined from its entropy.

math.ST↗
arXiv2015

Existence and Uniqueness of Proper Scoring Rules

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 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, Hyvärinen entropy, and quadratic entropy.

math.ST↗