Logarithmic score and information divergence appear in both information theory, statistics, statistical mechanics, and portfolio theory. We demonstrate that all these topics involve some kind of optimization that leads directly to the use of Bregman divergences. If a sufficiency condition is also fulfilled the Bregman divergence must be proportional to information divergence. The sufficiency condition has quite different consequences in the different areas of application, and often it is not fulfilled. Therefore the sufficiency condition can be used to explain when results from one area can be transferred directly from one area to another and when one will experience differences.
We give a new example for a proper scoring rule motivated by the form of Anderson--Darling distance of distribution functions and Example 5 in Brehmer and Gneiting (2020).
Averages of proper scoring rules are often used to rank probabilistic forecasts. In many cases, the individual terms in these averages are based on observations and forecasts from different distributions. We show that some of the most popular proper scoring rules, such as the continuous ranked probability score (CRPS), give more importance to observations with large uncertainty which can lead to unintuitive rankings. To describe this issue, we define the concept of local scale invariance for scoring rules. A new class of generalized proper kernel scoring rules is derived and as a member of thi…
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。
Many forecasts consist not of point predictions but concern the evolution of quantities. For example, a central bank might predict the interest rates during the next quarter, an epidemiologist might predict trajectories of infection rates, a clinician might predict the behaviour of medical markers over the next day, etc. The situation is further complicated since these forecasts sometimes only concern the approximate "shape of the future evolution" or "order of events". Formally, such forecasts can be seen as probability measures on spaces of equivalence classes of paths modulo time-parametriz…
We develop new approaches in multi-class settings for constructing proper scoring rules and hinge-like losses and establishing corresponding regret bounds with respect to the zero-one or cost-weighted classification loss. Our construction of losses involves deriving new inverse mappings from a concave generalized entropy to a loss through the use of a convex dissimilarity function related to the multi-distribution $f$-divergence. Moreover, we identify new classes of multi-class proper scoring rules, which also recover and reveal interesting relationships between various composite losses curren…
When Are Scoring Rules Proper? Bridging Theory and Practice in Survival Model Evaluation
Department of Cancer Genetics, Institute for Cancer Research, Oslo University Hospital（癌症遗传学系，癌症研究所，奥斯陆大学医院）；Department of Biostatistics, Faculty of Medicine, Oslo Centre for Biostatistics and Epidemiology (OCBE), University of Oslo, Norway（生物统计学系，医学学院，奥斯陆生物统计与流行病学中心（OCBE），奥斯陆大学，挪威）；OSPO Now, London, UK（伦敦，英国OSPO Now）；Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Oslo, Norway（数学系，数学与自然科学学院，奥斯陆大学，挪威）；Department of Statistics, LMU Munich, Germany（统计学系，慕尼黑大学）；Munich Center for Machine Learning (MCML), LMU Munich, Germany（慕尼黑机器学习中心（MCML），慕尼黑大学，德国）；Statistical Consulting Unit (StaBLab), LMU Munich, Germany（统计咨询单位（StaBLab），慕尼黑大学，德国）；Leibniz Institute for Prevention Research and Epidemiology - BIPS, Bremen, Germany（预防研究与流行病学研究所—BIPS，不莱梅，德国）；Faculty of Mathematics and Computer Science, University of Bremen, Germany（数学与计算机科学学院，不莱梅大学，德国）
