Entropy on covers

As a generalization of partitions, covers allow overlaps between their members. In this paper, we will propose a family of entropy-like measures over covers, which are anti monotonic with regard to the partial order defined by refinement relations of covers. In parallel to the entropy theory, we also develop their conditional forms, which in turn lead to a family of semi-metrics on covers. These make it possible to apply entropy-based techniques in data mining or machine learning to problems naturally modelled by covers.