精确性系统：Pre-Dynkin 系统上的相千概率与线性子空间上的相干预vis
Systems of Precision: Coherent Probabilities on Pre-Dynkin-Systems and Coherent Previsions on Linear Subspaces
AI总结：
本文研究不精确概率的精确性系统，证明其在温和条件下构成 pre-Dynkin 系统，并揭示相干性与可扩张性等价及格对偶，还推广到线性子空间上的部分期望。
摘要：
在不精确概率文献中，很少有人关注不精确概率在某一事件集合上是精确的这一事实。我们将这些集合称为精确性系统。我们证明，在温和假设下，下概率与上概率的精确性系统构成一个所谓的（pre-）Dynkin 系统。有趣的是，在从基于部分数据的机器学习、频率概率论到量子概率论以及不确定性下决策的若干情境中，先验上人们只希望概率在某个特定的底层集合系统上是精确的。在这里，（pre-）Dynkin 系统也被采用作为精确性系统。我们证明，在可扩张性条件下，这些配备了概率的 pre-Dynkin 系统可以嵌入到集合代数中。令人惊讶的是，量子概率中一系列工作所阐述的可扩张性条件，等价于不精确概率文献中的相干性。在此基础上，我们阐明了一种格对偶，它将精确性系统与概率的 credal 集联系起来。我们以将该框架推广到不精确概率的期望型对应物来结束论述。pre-Dynkin 系统的类似物被证明是有界实值函数空间中的线性子空间（之集合）。我们引入了部分期望，即定义在 pre-Dynkin 系统上的概率的自然推广。同样，相干性与可扩张性是等价的。一种相关但更一般的格对偶保留了精确性系统与概率的 credal 集之间的关系。
英文摘要：
In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin-systems turn out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin-systems. Again, coherence and extendability are equivalent. A related, but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities.
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