On Logit Weibull Manifold
Prosper Rosaire Mama Assandje, Joseph Dongho, Thomas Bouetou Bouetou
TL;DR
The work analyzes whether a potential function and a gradient flow can be defined on a Weibull statistical manifold. It shows that the base Weibull manifold lacks dual coordinates and a potential function, but introduces a logit Weibull model that restores these geometric structures through a potential Φ and yields a Hamiltonian, completely integrable gradient system. Using information-geometry concepts (Fisher metric, dual coordinates, Legendre duality), the paper provides an explicit potential-driven framework and gradient dynamics for the logit Weibull manifold, along with conditions ensuring Hamiltonian integrability. This approach offers a principled way to realize Liouville-integrable dynamics on Weibull-family models by embedding them in a logit-augmented geometry, bridging standard statistical manifolds with Hamiltonian gradient flows.
Abstract
In this work, it is shown that there is no potential function on the Weilbull statistical manifold. However, from the two-parameter Weibull model we can extract a model with a potential function called the logit model. On this logit model, there is a completely integrable Hamiltonian gradient system.