Information Geometry by Nihat Ay Jürgen Jost Hông Vân Lê & Lorenz Schwachhöfer

Author:Nihat Ay, Jürgen Jost, Hông Vân Lê & Lorenz Schwachhöfer
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(4.120)

Since the geometry is determined by the derivatives of at , we consider the case where and are close to each other, that is

(4.121)

is small for all . We evaluate the divergence by Taylor expansion up to . Note that is of order .

Proposition 4.1

When is small, the canonical divergence (4.109) is expanded as

(4.122)

where

(4.123)

Proof

The Taylor series expansion of the local coordinates of the geodesic is given by

(4.124)

where . This follows from , , and (see the geodesic equation (B.38) in Appendix B). When is small, is of order . Hence, we regard (4.124) as a Taylor expansion with respect to and when is small. When , we have

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