From Interval (Set) and Probabilistic Granules to Set-and-Probabilistic Granules of Higher Order

In this chapter, we provide a natural motivation for granules of higher order, and we show that these granules provide a unified description of different uncertainty formalisms such as random sets, Dempster-Shafer approach, fuzzy sets, imprecise probabilities, and Bayesian statistics. We also prove that for fuzzy uncertainty, granules of second order are sufficient. 1 From an Ideal Exact Description to Interval (Set) and Probabilistic Granules Idealized objects. To describe the physical world, we identify objects: elementary particles, atoms, molecules, solid bodied, stars, etc. An ideal object should be well-defined. For example, in a geographic description, when we define forests, lakes, rivers, etc., we should be able to determine the exact boundary between a lake and a river that flows into this lake, the exact boundary between a forest and a nearby grassy area, etc. An ideal object should also be reasonably stable with time. From this viewpoint, • a river is a reasonable geographic object, because its path does not change much for a long time, while • a puddle – which is often easily visible too – is not a reasonable geographic object, because it can disappear in the course of hours. Group objects. The above description applied to individual objects: we can talk about the height of an individual man, the speed with which the individual river flows, etc. Vladik Kreinovich University of Texas at El Paso, El Paso, TX 79968, USA, e-mail: [email protected]