On escort distributions, q-gaussians and Fisher information

Abstract : 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 constraints in the form of a q‐average. A central example is the q‐gaussian, which is a generalization of the standard gaussian distribution. In this contribution, we show that escort distributions emerge naturally as a maximum entropy trade‐off between the distribution p(x) and the uniform distribution. This setting may typically describe a phase transition between two states. But escort distributions also appear in the fields of multifractal analysis, quantization and coding with interesting consequences. For the problem of coding, we recall a source coding theorem by Campbell relating a generalized measure of length to the Rényi‐Tsallis entropy and exhibit the links with escort distributions together with pratical implications. That q‐gaussians arise from the maximization of Rényi‐Tsallis entropy subject to a q‐variance constraint is a known fact. We show here that the (squared) q‐gaussian also appear as a minimum of Fisher information subject to the same q‐variance constraint.
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Jean-François Bercher. On escort distributions, q-gaussians and Fisher information. 30th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Jul 2010, Chamonix, France. pp.208-215, ⟨10.1063/1.3573618⟩. ⟨hal-00766571⟩

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