Proper local scoring rules on discrete sample spaces. (English) Zbl 1246.62010

Dawid, A. Philip; Lauritzen, Steffen; Parry, Matthew

Ann. Stat. 40, No. 1, 593-608 (2012).

Summary: 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.

MSC:

62C99	Statistical decision theory
05C90	Applications of graph theory

Keywords:

concavity; entropy; Euler’s theorem; supergradient; homogeneous functionS

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