Abstract
Divergence functions play a relevant role in Information Geometry as they allow for the introduction of a Riemannian metric and a dual connection structure on a finite dimensional manifold of probability distributions. They also allow to define, in a canonical way, a symplectic structure on the square of the above manifold of probability distributions, a property that has received less attention in the literature until recent contributions. In this paper, we hint at a possible application: we study Lagrangian submanifolds of this symplectic structure and show that they are useful for describing the manifold of solutions of the Maximum Entropy principle.
Highlights
Information Geometry [1,2] provides a sound and fruitful framework for interpreting statistics using classical differential geometry notions [3]
Exponential families are prototypical examples of finite dimensional manifolds admitting a dually flat canonical structure defined by the canonical divergence, and they play a relevant role in information geometry and statistics [1,2]
Their importance is due to the fact that they represent the manifold of solutions of the variational problem associated to the Maximum Entropy Principle (MEP) with linear constraints ([11,12])
Summary
Information Geometry [1,2] provides a sound and fruitful framework for interpreting statistics using classical differential geometry notions [3]. Exponential families are prototypical examples of finite dimensional manifolds admitting a dually flat canonical structure defined by the canonical divergence, and they play a relevant role in information geometry and statistics [1,2] For our argument, their importance is due to the fact that they represent the manifold of solutions of the variational problem associated to the Maximum Entropy Principle (MEP) with linear constraints ([11,12]). In some applications to statistical mechanics, e.g., in the descriptions of phase transitions in Ising spin systems, MEP with nonlinear constraints is considered, see, e.g., in [13,14,15] In this case, the set of possible solutions has a richer structure, which is well captured by a Lagrangian submanifold of T ∗ M(h, k ).
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