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1999•Theory of Probability and Its ApplicationsRequires access

A Curious Example from Statistical Differential Geometry

G. Kallianpur, Y. T. Kim

Open publisher page 15 citations

Abstract

We consider an example of a family of probability measures on an infinite dimensional space which are mutually singular. Although the Fisher information metric and its variants are not available, it is shown that the parameter manifold has a natural differential structure that is non-Riemannian with nonzero curvature. It is also shown that there is no Riemannian metric compatible with the natural affine connection for which the curvature is not zero.

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What this paper is about

We consider an example of a family of probability measures on an infinite dimensional space which are mutually singular. Although the Fisher information metric and its variants are not available, it is shown that the parameter manifold has a natural differential structure that is non-Riemannian with nonzero curvature. It is also shown that there is no Riemannian metric compatible with the natural affine connection for which the curvature is not zero.

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OpenAlex reports 15 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

We consider an example of a family of probability measures on an infinite dimensional space which are mutually singular. Although the Fisher information metric and its variants are not available, it is shown that the parameter manifold has a natural differential structure that is non-Riemannian with nonzero curvature. It is also shown that there is no Riemannian metric compatible with the natural affine connection for which the curvature is not zero.

Key concepts: Mathematics, Statistical manifold, Information geometry, Affine connection, Differential geometry, Curvature, Metric (unit), Fisher information metric

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