Hellinger distance

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

:$H^2(P,Q)\; =\; frac\{1\}\{2\}displaystyle\; int\; left(sqrt\{frac\{dP\}\{dlambda\; -\; sqrt\{frac\{dQ\}\{dlambda\; ight)^2\; dlambda.$

Here, "dP" / "dλ" and "dQ" / "d"λ 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

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

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

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

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

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

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

Example

The Hellinger distance "H"("P", "Q") between two normal distributions $Psimmathcal\{N\}(x;mu\_1,sigma\_1^2)$ and $Qsimmathcal\{N\}(x;mu\_2,sigma\_2^2)$ is: $Hleft(P,\; Q\; ight)=sqrt\{frac\{1\}\{2\}-sqrt\{frac\{sigma\_1sigma\_2\}\{2left(sigma\_1^2+sigma\_2^2\; ight)\}e^\{-frac\{1\}\{2\}frac\{left(mu\_1-mu\_2\; ight)^2\}\{sigma\_1^2+sigma\_2^2$

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