Granularity as an optimal approach to uncertainty - a general mathematical idea with applications to sleep, consumption, traffic control, learning, etc

Traditional statistical and fuzzy approaches to describing uncertainty are continuous in the sense that we use a (potentially infinite) set of values from the interval [0,1] to characterize possible degrees of uncertainty. In reality, experts describe their degree of belief by using one of the finitely many words from natural language; in this sense, the actual description of expert uncertainty is granular. In this paper, we show that, in some reasonable sense, granularity is the optimal way of describing uncertainty. A similar mathematical idea explains similar "granularity" in such diverse areas as sleep, consumption, traffic control and learning.