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September, 1994 Preferred Point Geometry and the Local Differential Geometry of the Kullback-Leibler Divergence
Frank Critchley, Paul Marriott, Mark Salmon
Ann. Statist. 22(3): 1587-1602 (September, 1994). DOI: 10.1214/aos/1176325644


A new preferred point geometric structure for statistical analysis, closely related to Amari's $\alpha$-geometries, is introduced. The added preferred point structure is seen to resolve the problem that divergence measures do not obey the intutively natural axioms for a distance function as commonly used in geometry. Using this tool, two key results of Amari which connect geodesics and divergence functions are developed. The embedding properties of the Kullback-Leibler divergence are considered and a strong curvature condition is produced under which it agrees with a statistically natural (squared) preferred point geodesic distance. When this condition fails the choice of divergence may be crucial. Further, Amari's Pythagorean result is shown to generalise in the preferred point context.


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Frank Critchley. Paul Marriott. Mark Salmon. "Preferred Point Geometry and the Local Differential Geometry of the Kullback-Leibler Divergence." Ann. Statist. 22 (3) 1587 - 1602, September, 1994.


Published: September, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0821.62004
MathSciNet: MR1311991
Digital Object Identifier: 10.1214/aos/1176325644

Primary: 53B99
Secondary: 62F05, 62F12

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 3 • September, 1994
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