Annals of Statistics

Proper local scoring rules on discrete sample spaces

A. Philip Dawid, Steffen Lauritzen, and Matthew Parry

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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.

Article information

Ann. Statist., Volume 40, Number 1 (2012), 593-608.

First available in Project Euclid: 7 May 2012

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Zentralblatt MATH identifier

Primary: 62C99: None of the above, but in this section
Secondary: 62A99: None of the above, but in this section

Concavity entropy Euler’s theorem supergradient homogeneous function


Dawid, A. Philip; Lauritzen, Steffen; Parry, Matthew. Proper local scoring rules on discrete sample spaces. Ann. Statist. 40 (2012), no. 1, 593--608. doi:10.1214/12-AOS972.

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