INFORMATION GEOMETRY
Gaussian manifold ≅ Poincaré half-plane
P₁: μ₁=
σ₁=
P₂: μ₂=
σ₂=
Geodesic points
40
Fisher-Rao dist
—
KL(P₁‖P₂)
—
KL(P₂‖P₁)
—
Euclidean dist
—
Fisher metric on Gaussians:
ds² = (dμ²+2dσ²)/σ²
≡ Poincaré half-plane H²
(μ,σ) ≡ (x,y), σ>0
Geodesics: semicircles
centered on μ-axis (σ=0)
dist²=2ln²(σ₂/σ₁)+...
(Atkinson-Mitchell 1981)