Linear Functions to the Extended Reals
Linear Functions to the Extended Reals
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扩展实数的线性函数
DOI：
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Bo Waggoner
中科院分区：
文献类型：
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作者：
Bo Waggoner
该注释从RD {±∞}进行了调查，该功能满足了线性的公理，无论扩展值算术允许，它们具有在d上定义的非平凡结构，并且与有限的线性函数不同，它们需要ω（D2）参数唯一地识别他们可以捕获垂直的切线到题词：当它在它的有效领域，只有当它是“仿射扩展”功能的最高范围时，这些结果都适用在这里，当可以从给定的凸功能构建适当的评分规则时，尤其要研究。
This note investigates functions from Rd to R∪{±∞} that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on d, and unlike finite linear functions, they require Ω(d2) parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never −∞) is convex if and only if it has an extended-valued subgradient at every point in its effective domain, if and only if it is the supremum of a family of “affine extended” functions. These results are applied to the well-known characterization of proper scoring rules, for the finite-dimensional case: it is carefully and rigorously extended here to a more constructive form. In particular it is investigated when proper scoring rules can be constructed from a given convex function.