The usual way probability theory is used is that you posit an algebra of events. Each such event has a probability (that is, it is "measurable"). The assumption that the set of events that have a probability is an algebra means that if A and B are both events then "A and B" is also an event (the intersection also has a probability). What happens if you do not make that assumption? In this paper (by Rabanus Derr and Bob Williamson) we provide an answer: you recover the theory of imprecise probability! This builds an intriguing bridge between measure theory and notions from social science such as intersectionality. One conclusion is that measurability should not be construed as a mere technical annoyance; rather, it is a crucial part of how you choose to model the world.
Our paper on The Richness of Calibration has appeared in the proceedings of FaCCT 2023
We recently finished two papers on imprecise probabilities.
This shows a relationship between the set system of measurable events and imprecision of probabilities, and thus offers a novel way of generalising traditional probabilities that is potentially useful for a range of problems, including modelling "intersectionality"
This shows that one can develop and strictly frequentist semantics for imprecise probabilities, whereby upper previsions arise from the set of cluster points of relative frequencies. This means the theory is applicable to all sequences, not just stochastic ones. We also present a converse result that suggests that the theory is the "right" thing in the sense that every upper prevision can be derived from a (non-stochastic sequence). The proof of this fact is constructive.