What is Predictive distribution, proper scoring rule?

A predictive distribution is a forecast of the whole conditional distribution $F(\cdot\mid x)$ of the target. A scoring rule $S(F, y)$ gives a loss to a forecast $F$ when $y$ occurs; it is a proper scoring rule if its expected value under the true distribution $G$ is minimised by forecasting $F = G$, so that honesty is optimal (Gneiting and Raftery, 2007). The negative log-likelihood is proper; so are the two rules below.