The geometry of proper scoring rules
The geometry of proper scoring rules
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DOI：
10.1007/s10463-006-0099-8
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发表时间：
2007-03-01
影响因子：
1
通讯作者：
Dawid, A. P.
中科院分区：
文献类型：
--
作者：
Dawid, A. P.
决策问题是依据结果空间、行动空间和损失函数来定义的。从这些简单的要素出发，我们能够构建：恰当计分规则；熵函数；散度函数；黎曼度量；以及无偏估计方程。从抽象的角度来看，损失函数定义了结果空间和行动空间之间的一种对偶性，而一个分布与其贝叶斯行为之间的对应关系诱导出一种自对偶性。这些共同为结果空间上的分布族确定了一种“决策几何”。这使得许多标准的统计概念和性质得以推广。特别是我们定义并研究了广义指数族。分析了几个例子，包括一种广义的布雷格曼几何。
A decision problem is defined in terms of an outcome space, an action space and a loss function. Starting from these simple ingredients, we can construct: Proper Scoring Rule; Entropy Function; Divergence Function; Riemannian Metric; and Unbiased Estimating Equation. From an abstract viewpoint, the loss function defines a duality between the outcome and action spaces, while the correspondence between a distribution and its Bayes act induces a self-duality. Together these determine a "decision geometry" for the family of distributions on outcome space. This allows generalisation of many standard statistical concepts and properties. In particular we define and study generalised exponential families. Several examples are analysed, including a general Bregman geometry.