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On the geometry of generalized Gaussian distributions

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  • Andai, Attila

Abstract

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovric, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.

Suggested Citation

  • Andai, Attila, 2009. "On the geometry of generalized Gaussian distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 777-793, April.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:4:p:777-793
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    References listed on IDEAS

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    1. Calvo, Miquel & Oller, Josep M., 1990. "A distance between multivariate normal distributions based in an embedding into the siegel group," Journal of Multivariate Analysis, Elsevier, vol. 35(2), pages 223-242, November.
    2. Lovric, Miroslav & Min-Oo, Maung & Ruh, Ernst A., 2000. "Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 36-48, July.
    3. Zografos, K., 1999. "On Maximum Entropy Characterization of Pearson's Type II and VII Multivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 67-75, October.
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    Keywords

    94A17 53B21 Gaussian distribution Differential geometry;

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