Information geometry in a global setting
Information geometry is a branch of statistical methods that apply techniques of differential geometry to the study of a parameter space of a family of probability distributions of a statistical model which can be viewed as points on a Riemannian manifold. \par A mapping torus $T_A$ of a hyperbolic automorphism $A\in \mathrm{SL}(2;\mathbb Z)$ of the 2-torus $\mathbb T^2$ is called a $\mathrm{Sol}^3$-manifold. The suspension Anosov flow on $T_A$ induces a positive contact structure $\xi_+=\ker(\beta_++\beta_-)$ and a negative contact structure $\xi_-=\ker(\beta_+-\beta_-)$, where $\beta_+$ and $\beta_-$ are the 1-forms defining the Anosov foliations. They form a bi-contact structure on $T_A$. If the parameter space is the upper half-plane $\mathbb H=\{(m,s): s>0\}\subset\mathbb R^2$ with a certain non-Kähler metric and a non-metric connection, where $m$ denotes the mean and $s$ is the standard deviation, then the local structures are derived from a certain global function $D$ on $\mathbb H\times\mathbb H$ which is fundamental in information theory. On the other hand, the product $\mathbb H\times\mathbb H$ parametrizes abelian surfaces in moduli theory, that is, the quotient of the product $\mathbb H\times\mathbb H$ under the action of a Hilbert modular group is a non-compact singular complex surface which parametrizes the isomorphism classes of complex abelian surfaces with a real multiplication structure. Its topology is defined by describing each end of a Hilbert modular surface as a cusp, namely the neighborhood of $\infty$ of the quotient $M$ of $\mathbb H\times\mathbb H$ under a certain action of a semi-direct product $\mathbb Z\ltimes\mathbb Z^2$ of lattices of $\mathbb R$ and $\mathbb R^2$. From the contact topological point of view, $M\cup \{\infty\}$ is the cone with positive and negative symplectic structures whose base $N$ is a $\mathrm{Sol}^3$-manifold with the positive and negative contact structures $\xi_{\pm}$ associated to the suspension Anosov flow. \par In this paper, the author considers the information geometry of normal distributions so that it fits with the contact topology of a cusp. \par The first result of the paper describes the information geometry of $\mathbb H$ by introducing a self-correspondence $F\subset\mathbb H\times\mathbb H$ which identifies the convolution operation for the probability densities in each factor of $\mathbb H\times\mathbb H$ to the operation of Bayesian learning for those in the other factor. \par The second result states that the action of $\mathbb Z\ltimes\mathbb Z^2$ preserves the metric and connection on each factor of $\mathbb H\times\mathbb H$, the sum (resp. the difference) of the natural area forms descends to the positive (resp. negative) symplectic structure on the quotient $M$, the surface $F$ is a Lagrangian correspondence which descends to a densely immersed Lagrangian submanifold $L\subset N$ of $M$ with respect to the negative symplectic structure, and $L$ is decomposed into Legendrian submanifolds of $N$ with respect to the positive contact structure. \par Finally, the author shows that there exists a contact Hamiltonian flow on the universal cover $\widetilde N\subset\mathbb H\times\mathbb H$ of the $\mathrm{Sol}^3$-manifold $N\subset M$ such that (i) the surface $F\subset N$ is an invariant submanifold, (ii) the projection of the induced flow on $F$ to each factor of $\mathbb H\times\mathbb H$ is tangent to a foliation whose leaves are geodesics for the non-metric connection in the information geometry, and (iii) the iterations of convolutions along the foliation of the first factor and a certain Bayesian learning process along the foliation of the second factor are identified via the correspondence $F$.
scientific article; zbMATH DE number 7032359