To discuss the existence and uniqueness of proper scoring rules one needs to extend the associated entropy functions as sublinear functions to the conic hull of the prediction set. In some natural function spaces, such as the Lebesgue $L^p$-spaces over $\mathbb R^d$, the positive cones have empty interior. Entropy functions defined on such cones have only directional derivatives. Certain entropies may be further extended continuously to open cones in normed spaces containing signed densities. The extended densities are Gâteaux differentiable except on a negligible set and have everywhere continuous subgradients due to the supporting hyperplane theorem. We introduce the necessary framework from analysis and algebra that allows us to give an affirmative answer to the titular question of the paper. As a result of this, we give a formal sense in which entropy functions have uniquely associated proper scoring rules. We illustrate our framework by studying the derivatives and subgradients of the following three prototypical entropies: Shannon entropy, Hyvärinen entropy, and quadratic entropy.
Order-Sensitivity and Equivariance of Scoring Functions
The relative performance of competing point forecasts is usually measured in terms of loss or scoring functions. It is widely accepted that these scoring function should be strictly consistent in the sense that the expected score is minimized by the correctly specified forecast for a certain statistical functional such as the mean, median, or a certain risk measure. Thus, strict consistency opens the way to meaningful forecast comparison, but is also important in regression and M-estimation. Usually strictly consistent scoring functions for an elicitable functional are not unique. To give guid…
Generative Adversarial Nets for Robust Scatter Estimation: A Proper Scoring Rule Perspective
University of Chicago（芝加哥大学）；Hong Kong University of Science and Technology（香港科技大学）
稳健散布估计是统计学中的一项基本任务。Gao et al.（2018）最近关于鲁棒估计和生成对抗网络（GANs）之间的联系的发现表明，可以使用优化GANs的类似技术来计算类似深度的鲁棒估计器。在本文中，我们介绍了一个一般的学习通过分类框架的基础上适当的评分规则的概念。这个框架允许我们通过$f$的变分近似的透镜来理解矩阵深度函数和各种GANs-由适当的评分规则引起的发散。然后，我们提出了一类新的强大的散布估计在这个框架中仔细构造鉴别器与适当的神经网络结构。在Huber污染模型下，证明了这些估计量能达到极小极大散布率估计.我们的数值结果表明，在各种设置下对文献中的竞争对手的良好性能。
