Hellinger Square-Root Geometry

The Hellinger distance quantifies the similarity between two probability distributions. For Bernoulli distributions P = (p, 1-p) and Q = (q, 1-q), we can map each distribution to a nonnegative unit vector (√p, √(1-p)) and (√q, √(1-q)) on the positive quadrant of the unit circle. The normalized Hellinger distance represents the Euclidean distance (the chord) between these square-root vectors, scaled by 1/√2.

Worked example: Computing distances

Consider two Bernoulli distributions with P = (0.25, 0.75) and Q = (0.75, 0.25).

First, find their square-root vectors:
√P = (√0.25, √0.75) = (0.5, 0.866...)
√Q = (√0.75, √0.25) = (0.866..., 0.5)
The Bhattacharyya coefficient (BC) is the dot product of these vectors:
BC = √0.25 * √0.75 + √0.75 * √0.25 = 2 * (0.5 * 0.866...) ≈ 0.866...
The normalized squared Hellinger distance (H²) is defined as:
H² = 1 - BC ≈ 1 - 0.866... = 0.13397...
This normalizes the maximum possible squared distance to 1 (when distributions are disjoint).
The Total Variation (TV) distance for Bernoulli distributions is:
TV = |p - q| = |0.25 - 0.75| = 0.5

Notice the relationship H² ≤ TV ≤ H√(2 - H²), demonstrating how Hellinger distance provides bounds for Total Variation.

Explore the geometry

Open the Hellinger Square-Root Geometry experiment to visually and interactively compare distributions.

Select Probability P = 0.25 and Probability Q = 0.75. Observe the two vectors on the unit circle arc and the chord connecting them.
Change Probability P to 0.5. The chord shortens as the vectors move closer, and the Hellinger distance decreases.
Set Probability P = 0 and Probability Q = 1. The distributions are completely disjoint. Their square-root vectors lie on the axes, the Bhattacharyya coefficient is exactly 0, and the normalized Hellinger distance is exactly 1. The angle between them is exactly π/2 radians.

Limitations of this geometric view

While the mapping to the unit sphere is perfectly visualizable in 2D for Bernoulli distributions, categorical distributions with many outcomes or continuous distributions require higher-dimensional or infinite-dimensional Hilbert spaces. The geometric intuition of chords and angles still mathematically applies, but it cannot be easily drawn. Additionally, this module focuses strictly on the 1/√2 normalized Hellinger convention (where the maximum distance is 1), which differs from some literature that omits this scaling factor (where the maximum distance is √2).

Sources

The geometric interpretation of Hellinger distance as chords between square-root vectors is detailed in learning theory literature.

Quantum Learning Theory (Lecture 5 Scribe Notes) - Discusses the Hellinger distance and its relationships to Total Variation.

Hellinger Square-Root Geometry