Logarithmic scoring rule ($S(p,y)=\log p(y)$)

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.