Proper scoring rule

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A scoring rule is proper if the expected score is maximized when the forecaster reports their true subjective probability distribution.

For the binary case, suppose a forecaster believes the outcome will occur with probability <math>p</math> and reports belief <math>b</math>, which will be scored by the rule <math>f</math>. We call <math>f</math> proper if <math>E[f(b, x)]</math> is maximized when <math>b = p</math>. We call <math>f</math> strictly proper if <math>f</math> is proper and <math>E[f(b, x)]</math> is maximized only for <math>b = p</math>.

Examples