Part 6 of 15. A statistical manifold comes with not one "straight line" but three, and the right one depends on what you are trying to compute. Amari's α-connections are a one-parameter family that interpolates between them and exposes a deep duality between exponential and mixture geometries.
Play: Drag the endpoint Gaussians to see the three different ways a statistical manifold defines a "straight line". The green path is the Fisher/Levi-Civita geodesic (shortest distance), the red is the e-geodesic (exponentially straight), and the blue is the m-geodesic (moment-matching straight).
Explore: Move the slider to see what an m-geodesic actually means. The true m-geodesic between two distributions is a literal mixture (left), which immediately leaves the Gaussian family and becomes bimodal. The blue curve above is its projection back onto the closest moment-matching Gaussian (right).
A metric alone isn't enough to define geometry; you also need a connection—a rule for what "straight" means when moving between points. Amari's α-connections (1982) are a one-parameter family of connections:
The e- and m-connections are dual to each other with respect to the Fisher metric :
A statistical manifold is often dually flat, meaning it has zero curvature under both the e- and m-connections, even if it has negative curvature under the Levi-Civita connection (like the Gaussian family). These two flat coordinate systems are related by Legendre duality.