  We investigate proper scoring rules for continuous distributions on the real
line. It is known that the log score is the only such rule that depends on the
quoted density only through its value at the outcome that materializes. Here we
allow further dependence on a finite number $m$ of derivatives of the density
at the outcome, and describe a large class of such $m$-local proper scoring
rules: these exist for all even $m$ but no odd $m$. We further show that for
$m\geq2$ all such $m$-local rules can be computed without knowledge of the
normalizing constant of the distribution.
