Abstract

To extend the classical Shannon entropy to nonadditive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this new concept on the basis of three axioms: the symmetry property, a boundary condition for which the entropy reduces to the Shannon entropy, and a generalized version of the well-known recursivity property. We also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy-like measure. Lastly, we provide an interpretation of it in the framework of aggregation by the discrete Choquet integral.

Keywords: Entropy; Choquet capacity; Choquet integral; Information theory

Article Outline

1. Introduction

2. Uniformity of a discrete Choquet capacity

2.1. Notation and first definitions

2.2. Choquet capacities and maximal chains

2.3. Uniformity of a discrete Choquet capacity

3. Axiomatization of the entropy H_M

3.1. Additional definitions

3.2. Axioms

3.3. Axiomatic characterization of H_M

4. Properties of the entropy H_M

5. Interpretation of the entropy H_M in the aggregation framework

6. Conclusion

Appendix A. Proof of Theorem 2

References