Let $ S$ be a scoring rule for probability functions. Define

$ EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$ .

Say that $ S$ is striclty proper if and only if $ S$ minimises $ EXP_{S}(Q|P)$ as a function of $ Q$ . Define

$ D_{S}(P, Q) = EXP_{S}(Q|P) – EXP_{S}(P|P)$ .

If $ S$ is the logarithmic scoring rule defined by $ S(P, w) = -ln(P(w))$ , then $ D_{S}(P, Q)$ is just the Kullback-Leibler divergence between $ P$ and $ Q$ , or equivalently, the inverse Kullback-Leibler divergence between $ Q$ and $ P$ . Note that the inverse Kullback-Leibler divergence is an $ f$ -divergence.

My question is this: *is there any other strictly proper scoring rule $ S$ such that $ D_{S}(P, Q)$ is equal to $ F(Q, P)$ for some $ f$ -divergence $ F$ ?*

I think that $ D_{S}(P, Q)$ is always a Bregman divergence, and Amari proved that the only $ f$ -divergence that is also a Bregman divergence is the Kullback-Leibler divergence (on the space of probability functions). Is this enough to imply that there are no other strictly proper scoring rules with this property?