In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system 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. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. 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 physics are equivalent to coherence in the sense of Walley (1991, Statistical reasoning with imprecise probabilities, p. 84). Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities.

翻译：在关于不精确概率的文献中，很少注意到不精确概率在某些事件上是精确的这一事实。我们称这些集合为精度系统。我们证明，在温和假设下，下概率和上概率的精度系统构成一个所谓的（预）Dynkin系统。有趣的是，从部分数据上的机器学习、频率概率论到量子概率论以及不确定性下的决策制定等多个场景中，概率先验上只期望在特定的底层集合系统上保持精确。所有这些场景的核心在于观察到：两个事件上的精确信念、概率或频率并不必然意味着这些事件交集的精度也成立。此处，（预）Dynkin系统同样被用作精度系统。我们证明，在可扩展性条件下，这些带有概率的预Dynkin系统可以嵌入到集合代数中。令人惊讶的是，量子物理学研究中提炼出的可扩展性条件等价于Walley（1991，《不精确概率的统计推理》，第84页）意义上的相干性。因此，关于预Dynkin系统上概率的文献与不精确概率文献得以关联。最后，我们阐述了一种格对偶性，该对偶性严格地将精度系统与概率的credal集联系起来。特别地，我们提供了一个此前未被描述的参数化相干不精确概率族。