The Annals of Statistics
- Ann. Statist.
- Volume 40, Number 1 (2012), 593-608.
Proper local scoring rules on discrete sample spaces
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 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 . 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.
Ann. Statist., Volume 40, Number 1 (2012), 593-608.
First available in Project Euclid: 7 May 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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. https://projecteuclid.org/euclid.aos/1336396184