Comparing Dependencies in Probability Theory and General Rough Sets: Part-A

TL;DR

This paper explores the comparison of dependence concepts in probability theory and general rough sets, using duality and logic-based frameworks to clarify limitations and possibilities of translation between these methodologies.

Contribution

It introduces duality results to enhance comparison of dependence notions across probability and rough set theories within a unified logical framework.

Findings

Duality results demonstrate improved comparison capabilities.

Positive results show potential for cross-framework implications.

Negative results clarify the boundaries of possible translations.

Abstract

The problem of comparing concepts of dependence in general rough sets with those in probability theory had been initiated by the present author in some of her recent papers. This problem relates to the identification of the limitations of translating between the methodologies and possibilities in the identification of concepts. Comparison of ideas of dependence in the approaches had been attempted from a set-valuation based minimalist perspective by the present author. The deviant probability framework has been the result of such an approach. Other Bayesian reasoning perspectives (involving numeric valuations) and frequentist approaches are also known. In this research, duality results are adapted to demonstrate the possibility of improved comparisons across implications between ontologically distinct concepts in a common logic-based framework by the present author. Both positive and negative results are proved that delimit possible comparisons in a clearer way by her.