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 $$p$$ and reports belief $$b$$, which will be scored by the rule $$f$$. We call $$f$$ proper if $$E[f(b, x)]$$ is maximized when $$b = p$$. We call $$f$$ strictly proper if $$f$$ is proper and $$E[f(b, x)]$$ is maximized only for $$b = p$$.

Examples

 * Log score
 * Brier score
 * More on Wikipedia