Hellinger distance

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In probability theory, a branch of mathematics, given two probability measures P and Q that are absolutely continuous in respect to a third probability measure λ, the square of the Hellinger distance between P and Q is defined as the quantity

$H^2(P,Q) = \frac{1}{2}\displaystyle \int \left(\sqrt{\frac{dP}{d\lambda}} - \sqrt{\frac{dQ}{d\lambda}}\right)^2 d\lambda.$

Here, dP / dλ 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 in respect to which both P and Q are absolutely continuous.

For compactness, the above formula is often written as

$H^2(P,Q) = \frac{1}{2}\int \left(\sqrt{dP} - \sqrt{dQ}\right)^2.$

Some authors omit the factor 1/2 in front of the integral.

The Hellinger distance H(P, Q) thus defined satisfies the property

$0\le H(P,Q) \le 1.$

The Hellinger distance is related to the Bhattacharyya distance $B C (P, Q)$ as it can be defined as

$H(P,Q) = \frac{1}{2} \sqrt{(2-2 BC(P,Q))}.$

[edit] References

Yang, Grace Lo; Le Cam, Lucien M. (2000). Asymptotics in Statistics: Some Basic Concepts. Berlin: Springer. ISBN 0-387-95036-2.
Vaart, A. W. van der. Asymptotic Statistics (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge, UK: Cambridge University Press. ISBN 0-521-78450-6.
Pollard, David E. (2002). A user's guide to measure theoretic probability. Cambridge, UK: Cambridge University Press. ISBN 0-521-00289-3.