Weighted Relative Group Entropies and Associated Fisher Metrics

A large family of new [Formula: see text]-weighted group entropy functionals is defined and associated Fisher-like metrics are considered. All these notions are well-suited semi-Riemannian tools for the geometrization of entropy-related statistical models, where they may act as sensitive controlling invariants. The main result of the paper establishes a link between such a metric and a canonical one. A sufficient condition is found, in order that the two metrics be conformal (or homothetic). In particular, we recover a recent result, established for [Formula: see text] and for non-weighted relative group entropies. Our conformality condition is “universal”, in the sense that it does not depend on the group exponential.