Multi-granulation fuzzy probabilistic rough sets and their corresponding three-way decisions over two universes

This article introduces a general framework of multi-granulation fuzzy probabilistic roughsets (MG-FPRSs) models in multi-granulation fuzzy probabilistic approximation space over twouniverses. Four types of MG-FPRSs are established, by the four different conditional probabilitiesof fuzzy event. For different constraints on parameters, we obtain four kinds of each type MG-FPRSsover two universes. To find a suitable way of explaining and determining these parameters in eachkind of each type MG-FPRS, three-way decisions (3WDs) are studied based on Bayesian minimum-riskprocedure, i.e., the decision-theoretic rough set (DTRS) approach. The main contribution of this paperis twofold. One is to extend the fuzzy probabilistic rough set (FPRS) to MG-FPRS model over two universes.Another is to present an approach to select parameters in MG-FPRS modeling by using the process ofdecision-making under conditions of risk.