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2005.41: Universal connection and curvature for statistical manifold geometry

2005.41: Khadiga Arwini, L Del Riego and CTJ Dodson (2007) Universal connection and curvature for statistical manifold geometry. Houston Journal of Mathematics, 33 (1). pp. 145-161. ISSN 0362-1588

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Abstract

Statistical manifolds are representations of smooth families of probability density functions that allow differential geometric methods to be applied to problems in stochastic processes, mathematical statistics and information theory. It is common to have to consider a number of linear connections on a given statistical manifold and so it is important to know the corresponding universal connection and curvature; then all linear connections and their curvatures are pullbacks. An important class of statistical manifolds is that arising from the exponential families and one particular family is that of gamma distributions, which we showed recently to have important uniqueness properties in stochastic processes. Here we provide formulae for universal connections and curvatures on exponential families and give an explicit example for the manifold of gamma distributions.

Item Type:Article
Uncontrolled Keywords:Statistical manifold, exponential family, connection, universal connection, universal curvature, gamma 2-manifold, Freund 4-manifold, bivariate Gaussian 5-manifold
Subjects:MSC 2000 > 53 Differential geometry
MSC 2000 > 62 Statistics
MIMS number:2005.41
Deposited By:Prof CTJ Dodson
Deposited On:16 April 2007

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