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