PROPER LOCAL SCORING RULES ON DISCRETE SAMPLE SPACES

• 成果类型: Article
• 署名作者: Dawid, Philip; Lauritzen, Steffen; Parry, Matthew
• 署名单位: University of Cambridge; University of Oxford; University of Otago
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS972
• 发表日期: 2012
• 页码: 593-608
• 关键词: information

摘要

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 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 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 Hyvarinen's method of ratio matching.