Change the coefficients of conditional entropies in extensivity
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The Boltzmann–Gibbs entropy is a functional on the space of probability measures. When a state space is countable, one characterization of the Boltzmann–Gibbs entropy is given by the Shannon–Khinchin axioms, which consist of continuity, maximality, expandability and extensivity. Among these four properties, the extensivity is generalized in various ways. The extensivity of a functional is interpreted as the property that, for any random variables (X,Y) taking finitely many values in ℕ, the difference between the value of the functional at the joint law of (X,Y) and that at the law of X coincides with the linear combinations of the values at the conditional laws of Y given X=n with coefficients given by the probabilities of each event X=n. A generalization of the extensivity obtained by replacing the coefficients with a power of the probabilities of the events X=n provides a characterization of the Tsallis entropy. In this paper, we first prove the impossibility to replace the coefficients with a non-power function of the probabilities of the events X=n. Then we estimate the difference between the value at the joint law of (X,Y) and that at the law of X for a general functional.
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