Maximum Entropy of Random Permutation Set

TL;DR

This paper introduces the concept of maximum entropy for random permutation sets (RPS), providing analytical solutions and demonstrating its relation to Deng and Shannon entropies through numerical examples.

Contribution

It presents the first analytical derivation of the maximum entropy for RPS and explores its connection to existing entropy measures.

Findings

Maximum entropy RPS aligns with maximum Deng entropy.

Maximum entropy RPS reduces to maximum Shannon entropy when permutations are single elements.

Numerical examples validate the theoretical results.

Abstract

Recently, a new type of set, named as random permutation set (RPS), is proposed by considering all the permutations of elements in a certain set. For measuring the uncertainty of RPS, the entropy of RPS is presented. However, the maximum entropy principle of RPS entropy has not been discussed. To address this issue, in this paper, the maximum entropy of RPS is presented. The analytical solution for maximum entropy of RPS and its corresponding PMF condition are respectively proofed and discussed. Numerical examples are used to illustrate the maximum entropy RPS. The results show that the maximum entropy RPS is compatible with the maximum Deng entropy and the maximum Shannon entropy. When the order of the element in the permutation event is ignored, the maximum entropy of RPS will degenerate into the maximum Deng entropy. When each permutation event is limited to containing just one element, the maximum entropy of RPS will degenerate into the maximum Shannon entropy.