Scoring rule ($S(P,x)$)

A scoring rule assigns a reward or loss to an announced probability distribution and realized outcome. In the reward convention, a proper scoring rule makes truthful reporting maximize expected reward; the loss convention reverses the comparison.

For a binary outcome, rewarding the reported probability of the realized outcome gives expected score $1-q+(2q-1)p$ under genuine probability $q$. The optimal report is an endpoint whenever $q\ne 1/2$, so this is not a proper scoring rule.

Truthful reporting is an optimizer of expected reward for every admissible true distribution. Propriety matters because it aligns a forecaster's reward with an honest expression of uncertainty.

A proper scoring rule is strictly proper when truthful reporting is the unique optimizing distribution. For densities, equality is understood almost everywhere relative to the reference measure.

Under true density $q$, the advantage of truthful reporting is the Kullback-Leibler divergence $D_{\mathrm{KL}}(q\Vert p)$. Its nonnegativity and equality condition prove strict propriety when the expected log scores are well defined.

The probability chain rule makes a sum of sequential predictive log scores equal the log Bayesian model evidence when the forecasts are coherent Bayesian one-step predictions under proper priors. Differences of total scores are therefore log Bayes factors.