  A scoring rule is a loss function measuring the quality of a quoted
probability distribution $Q$ for a random variable $X$, in the light of the
realized outcome $x$ of $X$; it is proper if the expected score, under any
distribution $P$ for $X$, is minimized by quoting $Q=P$. Using the fact that
any differentiable proper scoring rule on a finite sample space ${\mathcal{X}}$
is the gradient of a concave homogeneous function, we consider when such a rule
can be local in the sense of depending only on the probabilities quoted for
points in a nominated neighborhood of $x$. Under mild conditions, we
characterize such a proper local scoring rule in terms of a collection of
homogeneous functions on the cliques of an undirected graph on the space
${\mathcal{X}}$. A useful property of such rules is that the quoted
distribution $Q$ need only be known up to a scale factor. Examples of the use
of such scoring rules include Besag's pseudo-likelihood and Hyv\"{a}rinen's
method of ratio matching.
