Derr and Williamson

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The Set Structure of Precision

In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We show that this system of precision forms, under mild assumptions, 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. 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. Thus, we link the literature on probabilities on pre-Dynkin-systems to the literature on imprecise probability. In fact, the system of precision and imprecise probabilities live in structural duality.