Ordered Continuous Means and Information

Abstract

Possibilistic measures of information have been characterized as a form of OWA operators on the possibility values. This correspondence can be carried on to the continuous domains and distributions.

Continuous information measures have been previously discussed only once in the open literature [12], while continuous OWA’s were only hinted at in conference papers [17, 18]. This chapter presents a method of defining such functions on a continuous universe of discourse — a domain which is a measurable space of measure 1. The method is based on the concept of rearangement [5] of a function, used in lieu of sorting for the discrete possibility values.

For a continuous distribution, represented by a measurable function f(x) on the domain of discourse X, first a decreasing rearrangement—f(x) on [0, μ(X)]—is constructed. Then, depending on μ(X), one of two definitions is appropriate

For technical reasons the quantification of uncertainty must be in the form of information distance [6, 13] measuring the departure from the most ‘uninformed’ distribution (constant possibility 1). The final form of the information content for possibility distribution f, defined on domain X, μ(X) = 1, is given by the continuous OWA operato

Relationship with the discrete OWA’s, and especially the discrete uncertainty measures, is discussed and various limit properties and approximations are established. Lastly, an investigation of continuous OWA as an integral transform is indicated.