On Monotone Embedding in Information Geometry

Jun Zhang Department of Psychology and Department of Mathematics, University of Michigan

Information Theory mathscidoc:1609.19005

Entropy, 17, (7), 4485-4499, 2015.6
A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari's \(\alpha\)-geometry through a general monotone embedding function. It will be pointed out here that this so-called \((F, G)\)-geometry (which includes \(F\)-geometry as a special case) is identical to Zhang's (2004) extension to the \(\alpha\)-geometry, where the name of the pair of monotone embedding functions \(\rho\) and \(\tau\) were used instead of \(F\) and \(H\) used in Harsha and Subrahamanian Moosath (2014). Their weighting function \(G\) for the Riemannian metric appears cosmetically due to a rewrite of the score function in log-representation as opposed to \((\rho, \tau)\)-representation in Zhang (2004). It is further shown here that the resulting metric and \(\alpha\)-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the \(\alpha\)-geometric structure. As a special case, Naudts' (2004) \(\phi\)-logarithm embedding (using the so-called \(\log_\phi\) function) is recovered with the identification \(\rho=\phi, \, \tau=\log_\phi\), with \(\phi\)-exponential \(\exp_\phi\) given by the associated convex function linking the two representations.
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  title={On Monotone Embedding in Information Geometry},
  author={Jun Zhang},
Jun Zhang. On Monotone Embedding in Information Geometry. 2015. Vol. 17. In Entropy. pp.4485-4499. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160920175654417627027.
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