Lifting statistical structures
arXiv:2203.02938 · doi:10.1142/S0129055X22500428
Abstract
We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric tensor is pseudo-Riemannian. What is more, natural lifts of potentials (called also divergence or contrast functions) turn out to be again potentials, this time for the lifted statistical structures. We propose an analogous procedure for lifting statistical structures on Lie algebroids and lifting contrast functions which are defined on Lie groupoids. In particular, we study in detail Lie groupoid structures of higher tangent bundles of Lie groupoids. Our geometric constructions of lifts are illustrated by explicit examples, including some important statistical models and potential functions on Lie groupoids.
arXiv comment44 pages, the version to be published in Rev. Math. Phys
References in corpus (16)
- Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics
- Higher vector bundles and multi-graded symplectic manifolds
- Tangent Lifts of Poisson and Related Structures
- Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
- Algebroids - general differential calculi on vector bundles
- Vector bundles over Lie groupoids and algebroids
- Graded bundles and homogeneity structures
- Variational calculus with constraints on general algebroids
- Linear duals of graded bundles and higher analogues of (Lie) algebroids
- Hamilton-Jacobi approach to Potential Functions in Information Geometry
- A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the - Family
- Lie groupoids in information geometry
- VB-structures and generalizations
- Information geometry on groupoids: the case of singular metrics
- Generalised Potential Functions in Differential Geometry and Information Geometry
- On the geometry of lift metrics and lift connections on the tangent bundle