论文标题
最低信息的适当评分规则
A proper scoring rule for minimum information copulas
论文作者
论文摘要
边缘分布均匀的多维分布称为Copulas。其中，满足对期望的限制，并且最接近Kullback-Leibler Divergence的独立分布，称为最小信息copula。最小信息的密度函数copula包含一组称为归一化函数的函数，通常很难计算。尽管提出了许多适当的评分规则，以实现标准化常数（例如指数族）的概率分布，但由于归一化功能，这些分数不适用于最低信息。在本文中，我们提出了条件的Kullback-Leibler评分，该评分避免了归一化函数的计算。其构建的主要思想是使用成对的观测值。我们表明，所提出的分数在副群密度函数的空间中严格适当，因此从其得出的估计器具有渐近一致性。此外，得分是相对于参数的凸，可以通过梯度方法轻松优化。
Multi-dimensional distributions whose marginal distributions are uniform are called copulas. Among them, the one that satisfies given constraints on expectation and is closest to the independent distribution in the sense of Kullback-Leibler divergence is called the minimum information copula. The density function of the minimum information copula contains a set of functions called the normalizing functions, which are often difficult to compute. Although a number of proper scoring rules for probability distributions having normalizing constants such as exponential families are proposed, these scores are not applicable to the minimum information copulas due to the normalizing functions. In this paper, we propose the conditional Kullback-Leibler score, which avoids computation of the normalizing functions. The main idea of its construction is to use pairs of observations. We show that the proposed score is strictly proper in the space of copula density functions and therefore the estimator derived from it has asymptotic consistency. Furthermore, the score is convex with respect to the parameters and can be easily optimized by the gradient methods.
