Non-Metricity in Information Geometry

The importance of non-metricity in information geometry is described through studies on gradient flow from the perspective of Weyl geometry. The Amari-Centsov tensor Ckij is the non-metricity tensor with respect to the α connection ∇(α) and Fisher metric gij. We derive the explicit expression of the α connection based on the non-metricity ∇k(α)gij=αCkij. The scalar field obtained from a potential function in information geometry plays a key role and characterizes Weyl's gauge field, Weyl's non-metricity, and the rate of the potential function during the associated gradient flow.