A note on curvature of α-connections of a statistical manifold

Author: Zhang, Jun

Source: Annals of the Institute of Statistical Mathematics, Volume 59, Number 1, March 2007 , pp. 161-170(10)

Publisher: Springer

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Abstract:

The family of α-connections ∇^(α) on a statistical manifold <EquationSource Format="TEX"><![CDATA[$$mathcal{M}$$]]></EquationSource> equipped with a pair of conjugate connections <EquationSource Format="TEX"><![CDATA[$$abla equiv abla^{(1)}$$]]></EquationSource> and <EquationSource Format="TEX"><![CDATA[$$abla^{ast} equiv abla^{(-1)}$$]]></EquationSource> is given as <EquationSource Format="TEX"><![CDATA[$$abla^{(alpha)}=frac{1+alpha}{2} abla + frac{1-alpha}{2} abla^{ast}$$]]></EquationSource> . Here, we develop an expression of curvature R ^(α) for ∇^(α) in relation to those for <EquationSource Format="TEX"><![CDATA[$$abla, abla^{ast}$$]]></EquationSource> . Immediately evident from it is that ∇^(α) is equiaffine for any <EquationSource Format="TEX"><![CDATA[$$alpha in mathbb{R}$$]]></EquationSource> when <EquationSource Format="TEX"><![CDATA[$$abla, abla^{ast}$$]]></EquationSource> are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011-1023, 2005). Other related formulae are also developed.

Keywords: Equiaffine connections; Parallel volume form; Ricci tensor; Cubic/skewness form

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10463-006-0105-1

Affiliations: 1: Email: junz@umich.edu

Publication date: 2007-03-01