A brief synopsis of progress in differential geometry in statistics is followed by a note\ud of some points of tension in the developing relationship between these disciplines. The preferred\ud point nature of much of statistics is described and suggests the adoption of a corresponding\ud geometry which reduces these tensions. Applications of preferred point geometry in statistics are\ud then reviewed. These include extensions of statistical manifolds, a statistical interpretation of\ud duality in Amari’s expected geometry, and removal of the apparent incompatibility between\ud (Kullback-Leibler) divergence and geodesic distance. Equivalences between a number of new\ud expected preferred point geometries are established and a new characterisation of total flatness\ud shown. A preferred point geometry of influence analysis is briefly indicated. Technical details are\ud kept to a minimum throughout to improve accessibility
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