Proper local scoring rules on discrete sample spaces

Philip Dawid; Steffen Lauritzen; Matthew Parry

doi:10.1214/12-AOS972

Proper local scoring rules on discrete sample spaces

Philip Dawid, Steffen Lauritzen, Matthew Parry

26 Citationer (Scopus)

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.

Originalsprog	Udefineret/Ukendt
Tidsskrift	Annals of Statistics
Vol/bind	40
Udgave nummer	1
Sider (fra-til)	593-608
Antal sider	16
ISSN	0090-5364
DOI	https://doi.org/10.1214/12-AOS972
Status	Udgivet - feb. 2012
Udgivet eksternt	Ja

Adgang til dokumentet

10.1214/12-AOS972

Citationsformater

@article{fa28efedd6c248038736321e992346a2,

title = "Proper local scoring rules on discrete sample spaces",

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 Hyv{\"a}rinen's method of ratio matching.",

author = "Philip Dawid and Steffen Lauritzen and Matthew Parry",

year = "2012",

month = feb,

doi = "10.1214/12-AOS972",

language = "Udefineret/Ukendt",

volume = "40",

pages = "593--608",

journal = "Annals of Statistics",

issn = "0090-5364",

publisher = "Institute of Mathematical Statistics",

number = "1",

}

TY - JOUR

T1 - Proper local scoring rules on discrete sample spaces

AU - Dawid, Philip

AU - Lauritzen, Steffen

AU - Parry, Matthew

PY - 2012/2

Y1 - 2012/2

N2 - 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.

AB - 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.

U2 - 10.1214/12-AOS972

DO - 10.1214/12-AOS972

M3 - Tidsskriftartikel

SN - 0090-5364

VL - 40

SP - 593

EP - 608

JO - Annals of Statistics

JF - Annals of Statistics

IS - 1

ER -