Dually flat geometries of the deformed exponential family

Harsha K.V. and Subrahamanian Moosath K.S.

Physica A: Statistical Mechanics and its Applications, 2015, vol. 433, issue C, 136-147

Abstract: An exponential family is dually flat with respect to Amari’s ±1 connection. A deformed exponential family which is a generalization of the exponential family has two dually flat structures called the U-geometry and the χ-geometry. In the case of an exponential family invariant α-geometry gives the dually flat structure. But for a deformed exponential family, one need to consider generalized geometric structures other than the invariant α-geometry. The (F,G)-geometry on a statistical manifold is such a generalized geometry defined using a general embedding function F and a positive smooth function G. In this paper, we present the role of the (F,G)-geometry in the study of a deformed exponential family. We show that the dually flat U-geometry is the (F,G)-geometry for suitable choices of F and G. Further we show that the dully flat χ-geometry is the conformal flattening of the (F,G)-geometry for suitable F and G.

Keywords: (F,G)-geometry; α-geometry; Deformed exponential family; Conformal flattening; Escort probability (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:433:y:2015:i:c:p:136-147

DOI: 10.1016/j.physa.2015.03.023

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