  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.
