← Information Geometry

Play: Drag the dot on the right manifold or adjust the sliders below to see how a point on the parameter manifold corresponds directly to a probability density curve. Explore how changing alters the distribution shape.

Explore: Observe that an identical coordinate step creates vastly different statistical changes depending on where you are on the manifold. Notice how KL divergence changes drastically across the three scenarios below despite the same parameter change.

Concept Summary

• Distributions are points. A parameterised statistical family is a manifold. For example, the Gaussian family is exactly the upper half-plane . The parameter values are just local coordinates (like latitude and longitude).
• Parameters are the wrong ruler. Euclidean distance between parameters does not reflect true statistical distance.
• Distance is distinguishability. The correct way to measure distance on this manifold is by how easily two distributions can be distinguished from their samples.