Figure 1.

Diagram of a statistical manifold. $\mathbf{\theta }$ and $s\left({\mathbf{\theta }}^{\prime }\right)$ denote parameters of the family of distributions from different samples $\mathit{X}$. The connection on the statistical manifold S represents a geodesic (the shortest line) between points $s\left(\mathbf{\theta }\right)$ and $s\left({\mathbf{\theta }}^{\prime }\right)$. The length of the geodesic serves as a distance measure between two points on the manifold. The arrow on the geodesic starting from the point $s\left(\mathbf{\theta }\right)$ denotes the tangent vector, which gives the direction of the geodesic.