Information Geometry of the Gaussian Space

TL;DR

This paper explores the geometric structure of Gaussian spaces using information geometry, focusing on entropy, translation continuity, inequalities, and differentiability of densities within the Pistone-Sempi exponential manifold framework.

Contribution

It provides a detailed analysis of the Gaussian space's geometric properties through the Pistone-Sempi exponential manifold, highlighting new insights into entropy and differentiability.

Findings

Characterization of the exponential manifold structure in Gaussian spaces

Analysis of entropy and translation continuity properties

Establishment of Poincaré-type inequalities and differentiability results

Abstract

We discuss the Pistone-Sempi exponential manifold on the finite-dimensional Gaussian space. We consider the role of the entropy, the continuity of translations, Poincar\'e-type inequalities, the generalized differentiability of probability densities of the Gaussian space.