Module fisher_rao

Fisher-Rao distance on the statistical manifold.

The Fisher-Rao metric is the unique Riemannian metric (up to scale) on the space of probability distributions that is invariant under sufficient statistics (Chentsov’s theorem). It is the mathematically correct way to measure distance between distributions.

For categorical distributions (region proportions), the Fisher-Rao distance has a closed-form solution via the Bhattacharyya angle:

$$d_{FR}(p, q) = 2 \arccos\left(\sum_i \sqrt{p_i \cdot q_i}\right)$$

Why Fisher-Rao for CVX

Region distributions at different timestamps are points on a statistical manifold. The Fisher-Rao distance is the geodesic distance on this manifold — more principled than L2, KL divergence, or even Wasserstein for measuring distributional change.

Comparison with Other Distances

Distance	Symmetric	Metric	Invariant	Bounded
KL divergence	No	No	Yes	No
L2	Yes	Yes	No	No
Wasserstein	Yes	Yes	No	No
Fisher-Rao	Yes	Yes	Yes	Yes [0, π]

References

• Rao, C.R. (1945). Information and accuracy attainable in estimation.
• Chentsov, N.N. (1982). Statistical Decision Rules and Optimal Inference.

Functions

bhattacharyya_coefficient

Bhattacharyya coefficient: BC(p, q) = Σ √(p_i × q_i).

fisher_rao_distance

Fisher-Rao distance between two categorical distributions.

fisher_rao_distance_f32

Fisher-Rao distance for f32 distributions (convenience wrapper).

hellinger_distance

Hellinger distance: H(p, q) = √(1 - BC(p, q)) / √2.