Information Geometry of the Gaussian Space

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, Poincaré-type inequalities, the generalized differentiability of probability densities of the Gaussian space.

Information Geometry of smooth densities on the Gaussian space: Poincaré inequalities

“…We have collected here a list of possible applications of the information geometry of the Gaussian space that has been introduced in [12,25] and further developed in the present paper.…”

“…Moreover, the closure of ∂ i is the infinitesimal generator of the translation operator. See the full theory in [15,3] and the applications to IG in [12,25].…”

“…We refer to [23,25] for the definition of maximal exponential manifold E (γ), and of statistical bundle S E (γ). Below we report the results that are necessary in the context of the present paper.…”

“…In the following proposition we give a summary of the inequalities proved so far. We do not care in this paper to define explicitly the relevant Gauss-Sobolev spaces, but this could be done in the spirit of [25]. Proposition 3.…”

“…In a series of papers [22,12,24,25], the idea of a non-parametric Information Geometry (IG), especially, IG of the Gaussian space has been explored. About the analysis of the Gaussian space, see, for example, [15,19].…”