Proper local scoring rules
Matthew Parry, A. Philip Dawid, Steffen Lauritzen
The Annals of Statistics · 2012 · Volume 40 · Issue 1
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 ≥ 2 all such m-local rules can be computed without knowledge of the normalizing constant of the distribution.
Keywords
62A99, 62C99, Bregman score, concavity, divergence, Entropy, Euler–Lagrange equation, homogeneity, integration by parts, local function, score matching, variational methods
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