Local proper scoring rules

  • Philip Dawid (University of Cambridge, Cambridge, United Kingdom)
  • Matthew Parry and Steffen Lauritzen
University n.n. Universität Leipzig (Leipzig)


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.


02.08.10 06.08.10

Information Geometry and its Applications III

Universität Leipzig (Leipzig) University n.n. University n.n.

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Nihat Ay

Max Planck Institute for Mathematics in the Sciences, Germany

Paolo Gibilisco

Università degli Studi di Roma "Tor Vergata", Italy

František Matúš

Academy of Sciences of the Czech Republic, Czech Republic