Aaron Mandel Lecture Series
Uncertainty and Information: Foundations and Applications of Generalized Information Theory
Dr. George J. Klir
Department of Systems Science & Industrial Engineering
Date:
Time:
Location: 004 Kalkin
A research program whose objective is to study uncertainty and uncertainty-based information in all their manifestations was introduced in the early 1990’s under the name “generalized information theory” (GIT). This research program, motivated primarily by some fundamental methodological issues emerging from the study of complex systems, is based on a two-dimensional expansion of classical, probability-based information theory. In one dimension, the theory of probability measures is expanded by abandoning the requirement of additivity; this leads to a theory of generalized measures. In the other dimension, the formalized language of classical set theory, within which probability measures are formalized, is expanded by abandoning the requirement that sets have sharp boundaries; this leads to theories of fuzzy sets of various types. As in classical information theory, uncertainty is the primary concept in GIT and information is defined in terms of uncertainty reduction. This restricted interpretation of the concept of information is described in GIT by the qualified term “uncertainty-based information.”
Each uncertainty theory that is recognizable within the expanded framework is characterized by: (i) a particular formalized language (a theory of fuzzy sets of some particular type); and (ii) a generalized measure of some particular type. In order to fully develop a particular uncertainty theory T requires that issues at each of the following four levels be adequately addressed: (i) an uncertainty function u of theory T must be formalized in terms of appropriate axioms; (ii) a calculus for dealing with function u must be developed; (iii) a justifiable uncertainty functional U must be determined to measure the amount of relevant uncertainty (predictive, retrodictive, prescriptive, diagnostic, etc.) in any situation formalizable in terms of function u; and (iv) methodological aspects of the theory must be developed by utilizing functional U as a measuring instrument.
Among the many uncertainty theories that are possible within the expanded conceptual framework, only a few theories (briefly examined in this presentation) have been sufficiently developed so far. One important result of research in the area of GIT is that the tremendous diversity of uncertainty theories made possible by the expanded framework is made tractable due to some key properties of these theories that are invariant across the whole spectrum. These unifying properties allows us to deal with uncertainty not only within a particular theory chosen in a given application context, but also, if desirable, within the whole spectrum of theories subsumed under GIT. That is, these properties allow us to move from one theory to another, as needed, when dealing with an application.
One of the unifying features of the various uncertainty theories is that two types of uncertainty coexist in each of them. These are usually referred to as nonspecificity and conflict. It is significant that well-justified measures of these two types of uncertainty are expressed by functionals of the same form in all the uncertainty theories, even though these functionals are subject to different calculi in different theories. Moreover, equations that express relationship between marginal, joint, and conditional measures of uncertainty are invariant across the whole spectrum of theories subsumed under GIT. The tremendous diversity of possible uncertainty theories is thus compensated by their many commonalities.
(This is a joint seminar with Civil & Environmental Engineering and Mathematics.)
Speaker bio:
George J. Klir is currently a
Distinguished Professor of Systems Science in Thomas J. Watson School of Engineering
and Applied Science. He received the M.S. Degree in Electrical Engineering from
the