Proper local scoring rules

TL;DR

This paper explores and characterizes a broad class of local proper scoring rules for continuous distributions, extending beyond the traditional log score by incorporating derivatives of the density at the outcome, with implications for scoring rule computation.

Contribution

It introduces a comprehensive class of m-local proper scoring rules for all even m, including rules that do not require normalization constants for m ≥ 2.

Findings

Existence of m-local proper scoring rules for all even m

No such rules exist for odd m

Rules for m ≥ 2 can be computed without the normalizing constant

Abstract

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.