Probabilistic rough set approximations

Probabilistic approaches have been applied to the theory of rough set in several forms, including decision-theoretic analysis, variable precision analysis, and information-theoretic analysis. Based on rough membership functions and rough inclusion functions, we revisit probabilistic rough set approximation operators and present a critical review of existing studies. Intuitively, they are defined based on a pair of thresholds representing the desired levels of precision. Formally, the Bayesian decision-theoretic analysis is adopted to provide a systematic method for determining the precision parameters by using more familiar notions of costs and risks. Results from existing studies are reviewed, synthesized and critically analyzed, and new results on the decision-theoretic rough set model are reported.