theorem hierarchical_kl {N : ℕ} (z : Fin N → ℝ) (hz : ∀ n, 0 < z n)
(treeCoeff : Fin N → ℝ) :
∑ n : Fin N, treeCoeff n * welfareDistanceFn (z n) =
∑ n : Fin N, treeCoeff n * klDivExp (1 / z n) 1 := by
congr 1; ext n
rw [welfare_eq_kl (hz n)]Information Geometry of CES: