# MathSciDoc: An Archive for Mathematician ∫

#### Information Theorymathscidoc: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.
@inproceedings{jun2015on,
title={On Monotone Embedding in Information Geometry},
author={Jun Zhang},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160920175654417627027},
booktitle={Entropy},
volume={17},
number={7},
pages={4485-4499},
year={2015},
}

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.