Abstract
Deng entropy has been proposed to measure the uncertainty degree of basic probability assignment in evidence theory. In this paper, the condition of the maximum of Deng entropy is discussed. According to the proposed theorem of the maximum Deng entropy, we obtain the analytic solution of the maximum Deng entropy, which yields that the most information volume of Deng entropy is bigger than that of the previous belief entropy functions. Some numerical examples are used to illustrate the basic probability assignment with the maximum Deng entropy.
Highlights
How to measure the uncertainty has attracted much attention [1]–[4]
Information entropy [17], derived from the Boltzmann-Gibbs (BG) entropy [20] in thermodynamics and statistical mechanics, has been an indicator to measures uncertainty which is associated with the probability density function (PDF)
In the classical entropy theory, assume a system described by variable x has N different states, i.e. xi, i = 1, . . . N, the classical maximum entropy of the system is log|N |
Summary
How to measure the uncertainty has attracted much attention [1]–[4]. A lot of theories have been developed, such as probability theory [5], fuzzy set theory [6], Dempster-Shafer evidence theory [7], [8], rough sets [9], generalized evidence theory [10] and D numbers [11]–[15]. A new entropy, Deng entropy is proposed to measure the uncertainty degree of basic probability assignment [26], which is possible to deal with the simultaneous events in the probable field [27], e.g. for the state {x1, x2}, the logic relation of x1, x2 is including AND. On the contrary, the correlations between the N elements are strong enough (a feature which might typically occur for nonergodic states, e.g., in Hamiltonian many body systems with long-range interactions, or which are strongly quantum entangled), the extensivity of entropy might be lost (at least at the level of a large subsystem of a much larger system), being incompatible with classical thermodynamics [19].
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