Hierarchical Structure of Uncertainty

Abstract

We introduce a new concept called uncertainty spaces which is an extended concept of probability spaces. Then, we express n-layer uncertainty which we call hierarchical uncertainty by a hierarchically constructed sequence of uncertainty spaces, called a U-sequence. We use U-sequences for providing examples that illustrate Ellsberg's paradox. We will use the category theory to get a bird's eye view of the hierarchical structure of uncertainty. We discuss maps between uncertainty spaces and maps between U-sequences, seeing that they form categories of uncertainty spaces and the category of U-sequences, respectively. We construct an endofunctor of the category of measurable spaces in order to embed a given U-sequence into it. Then, by the iterative application of the endofunctor, we construct the universal uncertainty space which may be able to serve as a basis for multi-layer uncertainty theory.