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February 2012 Proper local scoring rules on discrete sample spaces
A. Philip Dawid, Steffen Lauritzen, Matthew Parry
Ann. Statist. 40(1): 593-608 (February 2012). DOI: 10.1214/12-AOS972

Abstract

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 Hyvärinen’s method of ratio matching.

Citation

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A. Philip Dawid. Steffen Lauritzen. Matthew Parry. "Proper local scoring rules on discrete sample spaces." Ann. Statist. 40 (1) 593 - 608, February 2012. https://doi.org/10.1214/12-AOS972

Information

Published: February 2012
First available in Project Euclid: 7 May 2012

zbMATH: 1246.62010
MathSciNet: MR3014318
Digital Object Identifier: 10.1214/12-AOS972

Subjects:
Primary: 62C99
Secondary: 62A99

Keywords: concavity , Entropy , Euler’s theorem , homogeneous function , supergradient

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 1 • February 2012
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