评分函数是概率预报基于观测的回馈，若预报与观测更符合则给予更高分数。恰当性使得评分在观测的平均意义下，当预报与实际一致时达到极大。因而恰当的评分函数会鼓励决策者给出更合理的估计。本文主要讨论恰当评分这一函数类的结构性质及其在模型选择和似然估计中的拓展。凸集上的正则评分可由任意凸函数及其下梯度表示，此凸函数可等价表示为广义信息熵。为简化计算，恰当评分还可由负定核函数以估计的期望形式表出。此外基于分布或分位数的恰当评分也有着广泛的适用性。通过广义散度可将恰当评分应用于诸如交叉检验和贝叶斯因子等模型选择方法。经似然评分导出的最优评分估计作为极大似然形估计具有相应渐近性质。

Scoring rules are a class of functions which are the feedback based on the predictor and observations. Proper scoring rules make sure that the expected scores under the event attain maximum value if the forecast is consistent with the real distribution. Therefore, proper scores encourage decision makers to get a more reasonable estimation. This paper reviews the structure and some extension on model selection and likelihood inference of the aforementioned class of functions. A score is proper if and only if there exists a convex function and its subgradient satisfies a specific form. Furthermore, the convex function is related to information entropy. To simplify the complexity, proper scores can also be represented as negative definite kernels in the form of expectation under the predicted distribution. By defining the generalized divergence, proper scores will expand cross-validation and Bayes factor to wider areas. Using average score or likelihood score to assess parameters will constitute maximum likelihood type estimation.