We show that Tsallis’ distributions can be derived from the standard (Shannon) maximum entropy setting, by incorporating a constraint on the divergence between the distribution and another distribution imagined as its tail. In this setting, we find an underlying entropy which is the Rényi entropy. Furthermore, escort distributions and generalized means appear as a direct consequence of the cons...

Abstract: Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper, we consider a novel class of skew-normal mixture models, whose components capture skewnes...

Escort distributions are a simple one parameter deformation of an original distribution p. In Tsallis extended thermostatistics, the escort-averages, defined with respect to an escort distribution, have revealed useful in order to obtain analytical results and variational equations, with in particular the equilibrium distributions obtained as maxima of Rényi-Tsallis entropy subject to constrain...

Abstract—A random variable Z taking value in a finite, nonatomic measure space (X,M, μ) and whose distribution is absolutely continuous with respect to μ is to be described using N labels. We seek the labeling that minimizes the ρ-th moment of the μ-volume of the set of points in X that have the same label as Z. The large-N asymptotics of this minimum are expressed in terms of the Rényi entropy...

As additivity is a characteristic property of the classical information measure, Shannon entropy, pseudo-additivity of the form x+qy = x+y+(1−q)xy is a characteristic property of Tsallis entropy. Rényi in [1] generalized Shannon entropy by means of Kolmogorov-Nagumo averages, by imposing additivity as a constraint. In this paper we show that there exists no generalization for Tsallis entropy, b...

In this paper, we show that 1) additive energy is not appropriate for discussing the validity of Tsallis or Rényi statistics for nonextensive systems at equilibrium; 2) equilibrium N-body systems with nonad-ditive energy or entropy should be described by generalized statistics whose nature is prescribed by the existence of thermodynamic equilibrium. The nonextensive statistics may be Tsallis or...

We generalize the topological entanglement entropy to a family of topological Rényi entropies parametrized by a parameter alpha, in an attempt to find new invariants for distinguishing topologically ordered phases. We show that, surprisingly, all topological Rényi entropies are the same, independent of alpha for all nonchiral topological phases. This independence shows that topologically ordere...