From the Fisher Metric to the Symplectic Form: A Geometric Study of Dual Structures on Statistical Manifolds
Cauchy Mbakas'a Kongolo *
Faculty of Science and Technology, Department of Mathematics and Computer Science, National Pedagogical University (UPN), Kinshasa, Democratic Republic of Congo.
Patrick Tewo Nkinda
Faculty of Science and Technology, Department of Mathematics and Computer Science, National Pedagogical University (UPN), Kinshasa, Democratic Republic of Congo.
Arnold Mwambayi Kalonji
Faculty of Science and Technology, Department of Mathematics and Computer Science, National Pedagogical University (UPN), Kinshasa, Democratic Republic of Congo.
Jean-René Nteme Malama
Faculty of Science and Technology, Department of Mathematics and Computer Science, National Pedagogical University (UPN), Kinshasa, Democratic Republic of Congo.
Thierry Honorius Kabadi Kanza
Faculty of Science and Technology, Department of Mathematics and Computer Science, National Pedagogical University (UPN), Kinshasa, Democratic Republic of Congo.
Alain Musesa Landa
Faculty of Science and Technology, Department of Mathematics and Computer Science, University of Kinshasa, Kinshasa, Democratic Republic of Congo.
Gérard Tawaba Musian Ta-yen
Faculty of Science and Technology, Department of Mathematics and Computer Science, National Pedagogical University (UPN), Kinshasa, Democratic Republic of Congo.
*Author to whom correspondence should be addressed.
Abstract
This article studies the geometric relations between the Fisher metric, dual structures and the symplectic form of the cotangent bundle of a statistical manifold. We consider a regular statistical family endowed with the Fisher metric and, within a Hessian structure, a strictly convex potential \(\psi\). We show that the dual coordinates, defined by \(\eta_i=\frac{\partial \psi}{\partial \theta^i}\), are directly related to the Fisher metric by the relation:
\[
d \eta_i=\sum_j g_{i j} d \theta^j
\]
This relation expresses the local link between the Fisher metric and the dual coordinates: it allows this duality to be interpreted in the cotangent bundle \(T^* M\). We then consider the canonical symplectic form
\[
\omega=\sum_i d \theta^i \wedge d \eta_i
\]
and we prove that the graph of the differential of \(\psi\) is a Lagrangian submanifold. The results show that the same geometric potential enters the construction of the Fisher metric and of the dual coordinates, while its differential defines a Lagrangian submanifold of the cotangent bundle. We further show that the Fisher metric is obtained by restricting to this submanifold the canonical pairing between primal and dual coordinates. The study thus links, within a single mathematical construction, information geometry, statistical duality and symplectic geometry. Finally, we discuss the limitations of this construction, which remains local in nature and restricted to Hessian structures. The results are illustrated by the Bernoulli, Gaussian and multinomial families, the latter providing an example in arbitrary dimension n.
Keywords: Fisher metric, statistical manifold, dual structure, Hessian geometry, Legendre transform, symplectic form, Lagrangian submanifold