On Monotone Embedding in Information Geometry (Open Access)
MICHIGAN UNIV ANN ARBOR ANN ARBOR United States
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A paper was published Harsha and Subrahamanian Moosath, 2014 in which the authors claimed to have discovered an extension to Amaris 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 Zhangs 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 logphi function is recovered with the identification rhophi, taulogphi, with phi-exponential expphi given by the associated convex function linking the two representations.
- Theoretical Mathematics
- Statistics and Probability