Fuzzy probabilistic rough sets and their corresponding three-way decisions

The concept of probabilistic rough sets, as a main subject of this paper, is intimately connected with the concept of decision-theoretic rough sets. This paper investigates fuzzy and interval-valued fuzzy probabilistic rough sets within frameworks of fuzzy and interval-valued fuzzy probabilistic approximation spaces, respectively. Four types of fuzzy probabilistic rough sets as well as interval-valued fuzzy probabilistic rough sets are established in terms of different constraints on parameters. To find a suitable way of explaining and determining these parameters in each model, three-way decisions are studied based on Bayesian minimum risk decision procedure, i.e., the decision-theoretic rough set approach. The proposed models in this paper broaden applications of probabilistic rough sets due to their abilities of directly dealing with real-valued and interval-valued data.