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 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 Hyvärinen's method of ratio matching.
| Original language | Undefined/Unknown |
|---|---|
| Journal | Annals of Statistics |
| Volume | 40 |
| Issue number | 1 |
| Pages (from-to) | 593-608 |
| Number of pages | 16 |
| ISSN | 0090-5364 |
| DOIs | |
| Publication status | Published - Feb 2012 |
| Externally published | Yes |