Part 2 of 15. The right ruler for a statistical manifold exists, is unique, and (for the Gaussian family) makes the parameter plane look exactly like the Poincaré half-plane of hyperbolic geometry.
Play: Drag the blue point below to explore the Fisher ellipse field. Each ellipse is a local "unit sphere" of distinguishability. Notice how the ellipses shrink where information is high (near =0).
Explore: Compare the Fisher geodesic (purple) to the Euclidean straight line (amber). Drag either endpoint to see how the shortest path in parameter space arcs away from the boundary to minimize statistical distance.
The Fisher information metric provides the true ruler for a statistical manifold. By taking the gradient of the log-density , we get the score function:
The Fisher matrix is the covariance of the score:
Equivalently, it is the expected negative Hessian:
For the Gaussian family , this works out to a diagonal metric:
This metric defines a space of constant negative curvature, making the Gaussian manifold identical to the hyperbolic Poincaré half-plane.