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：在不精确概率文献中，鲜有关注到一个事实：不精确概率在某些事件上其实是精确的。我们称这些集合为精度系统。我们证明，在温和假设下，下概率与上概率的精度系统构成所谓的（前）Dynkin系统。有趣的是，在多个场景中——从基于部分数据的机器学习、频率概率论到量子概率论及不确定性下的决策——概率仅需在特定底层集合系统上保持精确。所有这些场景的核心观察在于：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的交集的精度也成立。在此，（前）Dynkin系统同样被采纳为精度系统。我们证明，在可延拓性条件下，这些配备概率的前Dynkin系统可嵌入集合代数。令人惊讶的是，量子物理学一系列工作中阐明的可延拓性条件，等价于Walley（1991，《不精确概率的统计推理》，第84页）意义上的协调性。由此，关于前Dynkin系统上概率的文献与不精确概率文献建立了联系。最后，我们阐明一种格对偶性，将精度系统与概率的信念集严格关联。特别地，我们提供了一个此前未被描述的、参数化的协调不精确概率族。