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Intrinsic Losses Based on Information Geometry and Their Applications

College of Mathematics, Sichuan University, Chengdu 610064, China
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Entropy 2017, 19(8), 405; https://doi.org/10.3390/e19080405
Received: 30 April 2017 / Revised: 21 July 2017 / Accepted: 3 August 2017 / Published: 6 August 2017
(This article belongs to the Special Issue Information Geometry II)
One main interest of information geometry is to study the properties of statistical models that do not depend on the coordinate systems or model parametrization; thus, it may serve as an analytic tool for intrinsic inference in statistics. In this paper, under the framework of Riemannian geometry and dual geometry, we revisit two commonly-used intrinsic losses which are respectively given by the squared Rao distance and the symmetrized Kullback–Leibler divergence (or Jeffreys divergence). For an exponential family endowed with the Fisher metric and α -connections, the two loss functions are uniformly described as the energy difference along an α -geodesic path, for some α { 1 , 0 , 1 } . Subsequently, the two intrinsic losses are utilized to develop Bayesian analyses of covariance matrix estimation and range-spread target detection. We provide an intrinsically unbiased covariance estimator, which is verified to be asymptotically efficient in terms of the intrinsic mean square error. The decision rules deduced by the intrinsic Bayesian criterion provide a geometrical justification for the constant false alarm rate detector based on generalized likelihood ratio principle. View Full-Text
Keywords: intrinsic loss; information geometry; exponential family; covariance matrix estimation; range-spread target detection intrinsic loss; information geometry; exponential family; covariance matrix estimation; range-spread target detection
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Rong, Y.; Tang, M.; Zhou, J. Intrinsic Losses Based on Information Geometry and Their Applications. Entropy 2017, 19, 405.

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