On Monotone Embedding in Information Geometry

Entropy

关键词: α-embedding;    monotone embedding;    conjugate embedding;    generalized Fisher-Rao metric;    Amari-Chentsov tensor;    deformed logarithm;    representation duality;    (ρ;    τ)-geometry;   

DOI  :  10.3390/e17074485

来源: mdpi

【 摘 要 】

A paper was published (Harsha and Subrahamanian Moosath, 2014) in which the authors claimed to have discovered an extension to Amari’s α-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 α-geometry, where the name of the pair of monotone embedding functions ρ and τ 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 (ρ, τ)-representation in Zhang (2004). It is further shown here that the resulting metric and α-connections obtained by Zhang (2004) through arbitrary monotone embeddings is a unique extension of the α-geometric structure. As a special case, Naudts’ (2004) ϕ-logarithm embedding (using the so-called log_ϕ function) is recovered with the identification ρ = ϕ, τ = log_ϕ, with ϕ-exponential exp_ϕ given by the associated convex function linking the two representations.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland