Two Bayesian approaches to rough sets

• We propose a unified framework to review, classify and examine fundamental issues of Bayesian approaches to rough sets.
• We identify two classes of Bayesian approaches to probabilistic rough sets and three fundamental issues.
• We explore a theory of three-way decisions as a tool for building ternary classifiers.
• We weave existing results into a coherent study of Bayesian approaches to rough sets.

Bayesian inference and probabilistic rough sets (PRSs) provide two methods for data analysis. Both of them use probabilities to express uncertainties and knowledge in data and to make inference about data. Many proposals have been made to combine Bayesian inference and rough sets. The main objective of this paper is to present a unified framework that enables us (a) to review and classify Bayesian approaches to rough sets, (b) to give proper perspectives of existing studies, and (c) to examine basic ingredients and fundamental issues of Bayesian approaches to rough sets. By reviewing existing studies, we identify two classes of Bayesian approaches to PRSsand three fundamental issues. One class is interpreted as Bayesian classification rough sets, which is built from decision-theoretic rough set (DTRS) models proposed by Yao, Wong and Lingras. The other class is interpreted as Bayesian confirmation rough sets, which is built from parameterized rough set models proposed by Greco, Matarazzo and Słowiński. Although the two classes share many similarities in terms of making use of Bayes’ theorem and a pair of thresholds to produce three regions, their semantic interpretations and, hence, intended applications are different. The three fundamental issues are the computation and interpretation of thresholds, the estimation of required conditional probabilities, and the application of derived three regions. DTRS models provide an interpretation and a method for computing a pair of thresholds according to Bayesian decision theory. Naive Bayesian rough set models give a practical technique for estimating probability based on Bayes’ theorem and inference. Finally, a theory of three-way decisions offers a tool for building ternary classifiers. The main contribution of the paper lies in weaving together existing results into a coherent study of Bayesian approaches to rough sets, rather than introducing new specific results.