Part 3 of 15. The Fisher metric is not one choice among many. It is the only Riemannian metric on a statistical manifold that is invariant under information-preserving transformations of the data, and therefore the only one measuring distributions rather than their labels.
Play: Click the split buttons to apply an information-preserving transformation (a Markov morphism) that redistributes probability mass. Observe that the Fisher-Rao distance remains perfectly invariant, while the Euclidean distance shrinks.
Explore: Drag the reference point around the 2-simplex to see Fisher-Rao isodistance contours. Notice how the contours compress near the corners (where certainty is high and distinguishability is easy) and dilate near the center.
Chentsov's theorem (1972) states that on the manifold of probability distributions, the Fisher information metric is the unique Riemannian metric (up to a scalar factor) that is invariant under Markov morphisms (information-preserving stochastic transformations).
A Markov morphism is a conditional distribution that maps a distribution on to on :
Because statistical distance must measure true distinguishability, it cannot depend on arbitrary outcome labels or uninformative splitting of states. For example, on the simplex, Euclidean distance fails this invariance test, while the Fisher-Rao distance succeeds: