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Abstract: Kullback–Leibler relative-entropy has unique properties in cases involving distributions resulting from relative-entropy minimization. Tsallis relative-entropy is a one-parameter generalization of Kullback–Leibler relative-entropy in the nonextensive thermostatistics. In this paper, we present the properties of Tsallis relative-entropy minimization and present some differences with the classical case. In the representation of such a minimum relative-entropy distribution, we highlight the use of the q-product, an operator that has been recently introduced to derive the mathematical structure behind the Tsallis statistics. One of our main results is the generalization of triangle equality of relative-entropy minimization to the nonextensive case. [Copyright &y& Elsevier]

Abstract: Shannon entropy of a probability measure P, defined as on a measure space , is not a natural extension from the discrete case. However, maximum entropy (ME) prescriptions of Shannon entropy functional in the measure-theoretic case are consistent with those for the discrete case. Also it is well known that Kullback–Leibler relative entropy can be extended naturally to measure-theoretic case. In this paper, we study the measure-theoretic aspects of nonextensive (Tsallis) entropy functionals and discuss the ME prescriptions. We present two results in this regard: (i) we prove that, as in the case of classical relative-entropy, the measure-theoretic definition of Tsallis relative-entropy is a natural extension of its discrete case, and (ii) we show that ME-prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case with respect to a particular instance of ME. [Copyright &y& Elsevier]