Power divergences on a probability simplex

Three probabilities, two-dimensional geometry

A family of discrepancies on one simplex

The triangle contains every q = (q₁, q₂, q₃) with qᵢ ≥ 0 and q₁ + q₂ + q₃ = 1. The red dot is the target p. Change the generator, or move the target to see the geometry change.

Generator power λ 1 · KL0: reverse KL · ½: 2 Hellinger² · 1: KL · 2: ½ Pearson χ² Target p₁ 0.160The third coordinate adjusts to keep total mass one. Target p₂ 0.580The available range changes with p₁.

Click or drag inside the triangle. Arrow keys move p₁/p₂; Shift takes larger steps; Home restores the target. Targets are projected to pᵢ ≥ 0.04, without modifying q. The color scale is fixed and saturates at 4; infinite boundary values use the darkest color. White contours are sampled at 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, and 4, omitting cells with an infinite corner. This is an f-divergence plot, not the Bregman divergence of the same scalar generator.