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,很少注意某些事件的准确概率是不准确的这一事实。我们称之为这些数据集的精确系统。我们表明,在轻度假设下,低和高概率的精确系统形成所谓的(Dynkin)系统。有趣的是,有几种环境,从机器学习关于超常概率理论的部分数据到量概率理论和在不确定性下的决策,先验性概率理论只希望精确到特定基本系统。所有这些环境的核心在于观察到,精确的信念、概率或两次事件的频率并不一定意味着保持这种精确度,以保持这些事件的交叉点。在这里,(Pre-Dynkin系统也被采用为精确度系统。我们表明,在可扩展性条件下,那些具备概率的先验性系统可以嵌入一个精确度的比值。令人惊讶的是,在量物理工作的一部分中阐述的准确性条件的准确性与准确性(Balley-Brilli)的准确性(我们从精确性的角度,统计性推导出准确性与精确性系统之间的精确性。