Probabilities, dependence and rough membership functions

Abstract

The goal of this research is to study the reflection of probability theories through rough membership functions in rough sets. Towards this, philosophy and variants of probability theories and their theorized connections with rough memberships functions are critically analyzed. The concept of rough membership functions and rough dependence are also generalized to granular operator spaces, characterized and used for the same purpose in more general contexts. A new theory of dependence based deviant probability is developed as a severe extension of the axiomatic approach to a dependence-based probability introduced recently by the present author. This permits clearer comparison of the concept with those of rough dependence. It is shown that the theories are very distinct semantically and similarities are poorly justified. These are relevant for rethinking the various probabilistic rough theories and related methodologies. The problem of contamination reduction was proposed recently across many papers by the present author. In this study, the scope of the problem within rough membership functions, probabilistic rough sets (PSTs) and three-way decision-making is also clarified by her. A new definition of artificial intelligence applicable in rough perspectives is also proposed on the basis of recent advances in algebraic semantics related to rough membership functions.