Conformal Flattening for Deformed Information Geometries on the Probability Simplex

Entropy (Basel). 2018 Mar 10;20(3):186. doi: 10.3390/e20030186.


Recent progress of theories and applications regarding statistical models with generalized exponential functions in statistical science is giving an impact on the movement to deform the standard structure of information geometry. For this purpose, various representing functions are playing central roles. In this paper, we consider two important notions in information geometry, i.e., invariance and dual flatness, from a viewpoint of representing functions. We first characterize a pair of representing functions that realizes the invariant geometry by solving a system of ordinary differential equations. Next, by proposing a new transformation technique, i.e., conformal flattening, we construct dually flat geometries from a certain class of non-flat geometries. Finally, we apply the results to demonstrate several properties of gradient flows on the probability simplex.

Keywords: Legendre conjugate; affine immersion; dually flat structure; gradient flow; invariance; nonextensive statistical physics; representing functions.