A scoring rule <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$S(x; q)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> provides a way of judging the quality of a quoted probability density <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>q</mi> </math> </EquationSource> </InlineEquation> for a random variable <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation> in the light of its outcome <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$x$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>x</mi> </math> </EquationSource> </InlineEquation>. It is called proper if honesty is your best policy, i.e., when you believe <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation> has density <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation>, your expected score is optimised by the choice <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$q=p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>q</mi> <mo>=</mo> <mi>p</mi> </mrow> </math> </EquationSource> </InlineEquation>. The most celebrated proper scoring rule is the logarithmic score, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$S(x; q)=-\log {q(x)}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>-</mo> <mo>log</mo> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>: this is the only proper scoring rule that is local, in the sense of depending on the density function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>q</mi> </math> </EquationSource> </InlineEquation> only through its value at the observed value <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$x$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>x</mi> </math> </EquationSource> </InlineEquation>. It is closely connected with likelihood inference, with communication theory, and with minimum description length model selection. However, every statistical decision problem induces a proper scoring rule, so there is a very wide variety of these. Many of them have additional interesting structure and properties. At a theoretical level, any proper scoring rule can be used as a foundational basis for the theory of subjective probability. At an applied level a proper scoring can be used to compare and improve probability forecasts, and, in a parametric setting, as an alternative tool for inference. In this article we give an overview of some uses of proper scoring rules in statistical inference, including frequentist estimation theory and Bayesian model selection with improper priors. Copyright Sapienza Università di Roma 2014
