On warped product on information geometry: statistical manifolds and statistical models

TL;DR

This paper investigates the application of warped product constructions within information geometry, analyzing their effects on divergences and geometric structures, and highlighting limitations in preserving canonical divergences for information-theoretic models.

Contribution

It introduces the warped product in the context of information geometry and examines its impact on geometric structures and divergences, revealing limitations in its applicability.

Findings

Warped product preserves certain geometric structures.

Canonical divergences are not preserved under warped product.

Warped product may lack relevance in information theory applications.

Abstract

We study a differential geometric construction, the warped product, on the background geometry for information theory. Divergences, dual structures and symmetric 3-tensor are studied under this construction, and we show that warped product of manifolds endow with such structure also is endowed with the same geometric notion. However, warped product does not preserve canonical divergences, which in particular shows that warped product lacks of meaning in the information theory setting.