Rough set theory offers new perspective on Bayes' theorem. The look on Bayes' theorem offered by rough set theory reveals that any data set (decision table) satisfies the total probability theorem and Bayes' theorem. These properties can be used directly to draw conclusions from objective data without referring to subjective prior knowledge and its revision if new evidence is available. Thus, the rough set view on Bayes' theorem is rather objective in contrast to subjective “classical” interpretation of the theorem. © 2003 Wiley Periodicals, Inc.

In this paper we study the granular structure of data in the language of rough set theory. It is shown that the granularity of data can be represented in a form of a flow graph, and the relationship between granules obeys Bayes' theorem. This leads to a new method of data analysis. This paper is dedicated to the renowned Sherlock Holmes for his mastery in theorizing about data.

This article introduces an approach to measures of information granules based on rough set theory. Informally, an information granule is a representation of a multiset (or bag) of real-world objects that are somehow indistinguishable, or similar, or which cause the same functionality. Examples of measures of information granules based on the rough set theory are inclusion, closeness, size, and enclosure. All of these measures are based on rough inclusion. This paper is limited to a consideration of measures of inclusion based on a straightforward extension of classical rough membership functions and closeness based on measurement of separation of equivalence classes in a partition of the universe containing information granules. Measurement of sensor-based information granules has been motivated by recent studies of sensor signals. A sensor signal is a non-empty, finite set of sample sensor signal values temporally ordered. Classification of sensor signals requires measurements of sample signal values over subintervals of time. This article introduces a rough set approach to measuring information granule inclusion and closeness.

In rough set theory with every decision rule two conditional probabilities, called certainty and coverage factors, are associated. These two factors are closely related with the lower and the upper approximation of a set, basic notions of rough set theory. It is shown that these two factors satisfy the Bayes' rule. The Bayes' rule in our case simply shows some relationship in the data, without referring to prior and posterior probabilities intrinsically associated with Bayesian inference. This relationship can be used to “invert” decision rules, i.e., to find reasons (explanation) for decisions thus providing inductive as well as deductive inference in our scheme.

In the rough set theory with every decision rule two conditional probabilities, called certainty and coverage factors, are associated. These two factors are closely related with the lower and the upper approximation of a set, basic notions of rough set theory. It is shown that these two factors satisfy the Bayes' theorem. The Bayes' theorem in our case simply shows some relationship in the data, without referring to prior and posterior probabilities intrinsically associated with Bayesian inference in our case and can be used to “inverse” decision rules, i.e., to find reasons (explanation) for decisions. © 2001 John Wiley & Sons, Inc.

Application of intelligent methods in industry become a very challenging issue nowadays and will be of extreme importance in the future. Intelligent methods include fuzzy sets, neural networks, genetics algorithms, and other techniques known as soft computing. No doubt, rough set theory can also contribute essentially to this domain. In this paper, basic ideas of rough set theory are presented and some possible intelligent industrial applications outlined.

This paper gives basic ideas of rough set theory a new approach to data analysis. The lower and upper approximation of a set, the basic operations of the theory, are intuitively explained and formally defined. Some applications of rough set theory are briefly outlined and some future problems are outlined.

The paper explores the concepts of approximate relations and functions in the framework of the theory of rough sets. The difficulties with the application of the idea of rough relation to general rough function definition are discussed. The definition of rough function for the domain of real numbers is introduced and its properties are investigated in detail including the generalization of the standard notion of function continuity known in the theory of real functions.

Vagueness and uncertainty have attracted the attention of philosophers and logicians for many years. Recently, AI researchers contributed essentially to this area of research. Fuzzy set theory and the theory of evidence are seemingly the most appealing topics. On this note we present a new approach, based on the rough set theory, for looking to these problems. The theory of rough sets seems a suitable mathematical tool for dealing with problems of vagueness and uncertainty. This paper is a modified version of the author's lecture titled “An inquiry into vagueness and uncertainty,” which was delivered at the AI Conference in Wigry (Poland), 1994.