Hellinger Distance - Definition - Measure Theory

Measure Theory

To define the Hellinger distance in terms of measure theory, let P and Q denote two probability measures that are absolutely continuous with respect to a third probability measure λ. The square of the Hellinger distance between P and Q is defined as the quantity

Here, dP / and dQ / dλ are the Radon–Nikodym derivatives of P and Q respectively. This definition does not depend on λ, so the Hellinger distance between P and Q does not change if λ is replaced with a different probability measure with respect to which both P and Q are absolutely continuous. For compactness, the above formula is often written as

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