On Monotone Embedding in Information Geometry (Open Access)
Abstract
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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 25, 2015
- Accession Number
- AD1042485
Entities
People
- Jun Zhang
Organizations
- University of Michigan