Entropy 2013, 15(12), 5384-5418; doi:10.3390/e15125384
Article

Nonparametric Information Geometry: From Divergence Function to Referential-Representational Biduality on Statistical Manifolds

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Received: 3 July 2013; in revised form: 11 October 2013 / Accepted: 22 October 2013 / Published: 4 December 2013
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, (Χ,μ). Classical information geometry prescribes, on Μθ: (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions ( α-divergence) defined on ΜθΜθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions) to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function (“referential duality”) and the other concerning its representation under an arbitrary monotone scaling (“representational duality”).
Keywords: Fisher information; alpha-connection; infinite-dimensional manifold; convex function
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MDPI and ACS Style

Zhang, J. Nonparametric Information Geometry: From Divergence Function to Referential-Representational Biduality on Statistical Manifolds. Entropy 2013, 15, 5384-5418.

AMA Style

Zhang J. Nonparametric Information Geometry: From Divergence Function to Referential-Representational Biduality on Statistical Manifolds. Entropy. 2013; 15(12):5384-5418.

Chicago/Turabian Style

Zhang, Jun. 2013. "Nonparametric Information Geometry: From Divergence Function to Referential-Representational Biduality on Statistical Manifolds." Entropy 15, no. 12: 5384-5418.

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