Bayes’ Theorem Revised — The Rough Set View

The look on Bayes' theorem offered by rough set theory is completely different from that used in the Bayesian data analysis philosophy and can be used directly to draw conclusions from data without referring to prior knowledge and its revision if new evidence is available.

Abstract

Rough set theory offers new insight into Bayes' theorem. The look on Bayes' theorem offered by rough set theory is completely different from that used in the Bayesian data analysis philosophy. It does not refer either to prior or posterior probabilities, inherently associated with Bayesian reasoning, but it reveals some probabilistic structure of the data being analyzed. It states that any data set (decision table) satisfies total probability theorem and Bayes' theorem. This property can be used directly to draw conclusions from data without referring to prior knowledge and its revision if new evidence is available. Thus in the presented approach the only source of knowledge is the data and there is no need to assume that there is any prior knowledge besides the data. We simply look what the data are telling us. Consequently we do not refer to any prior knowledge which is updated after receiving some data.