Local proper scoring rules

A scoring rule $S(x, Q)$ measures the quality of a quoted distribution $Q$ for an uncertain quantity $X$ in the light of the realised value $x$ of $X$. It is proper when it encourages honesty, i.e, when, if your uncertainty about $X$ is represented by a distribution $P$, the choice $Q = P$ minimises your expected loss. Traditionally, a scoring rule has been called local if it depends on $Q$ only through $q(x)$, the density of $Q$ at $x$. The only proper local scoring rule is then the $\log$-score, $-\log q(x)$. For the continuous case, we can weaken the definition of locality to allow dependence on a finite number m of derivatives of $q$ at $x$. A full characterisation is given of such order-$m$ local proper scoring rules, and their behaviour under transformations of the outcome space. In particular, any $m$-local scoring rule with $m > 0$ can be computed without knowledge of the normalising constant of the density. Parallel results for discrete spaces will be given.