Proper scoring rule

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>. <math>f</math> is proper if <math>E[f(b, x)]</math> is maximized when <math>b = p</math>