Hellinger distance

In probability and statistics, the Hellinger distance is used to quantify the similarity between two probability distributions. It is a type of f-divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by Ernst Hellinger.

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

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

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 with respect to which both P and Q are absolutely continuous. For compactness, the above formula is often written as

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

To define the Hellinger distance in terms of elementary probability theory, we take λ to be Lebesgue measure, so that dP / dλ and dQ / dλ are simply probability density functions. If we denote the densities as f and g, respectively, the squared Hellinger distance can be expressed as a standard calculus integral

$\frac{1}{2}\int \left(\sqrt{f(x)} - \sqrt{g(x)}\right)^2 dx = 1 - \int \sqrt{f(x) g(x)} \, dx,$

where the second form can be obtained by expanding the square and using the fact that the integral of a probability density over its domain must be one.

The Hellinger distance H(P, Q) satisfies the property (derivable from the Cauchy-Schwarz inequality)

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

The maximum distance 1 is achieved when P assigns probability zero to every set to which Q assigns a positive probability, and vice versa.

Sometimes the factor 1/2 in front of the integral is omitted, in which case the Hellinger distance ranges from zero to the square root of two.

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

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

Examples

The squared Hellinger distance between two normal distributions $\scriptstyle P\,\sim\,\mathcal{N}(\mu_1,\sigma_1^2)$ and $\scriptstyle Q\,\sim\,\mathcal{N}(\mu_2,\sigma_2^2)$ is:

$H^2(P, Q) = 1 - \sqrt{\frac{2\sigma_1\sigma_2}{\sigma_1^2+\sigma_2^2}} \, e^{-\frac{1}{4}\frac{(\mu_1-\mu_2)^2}{\sigma_1^2+\sigma_2^2}} .$

The squared Hellinger distance between two exponential distributions $\scriptstyle P\,\sim \,\rm{Exp}(\alpha)$ and $\scriptstyle Q\,\sim\,\rm{Exp}(\beta)$ is:

$H^2(P, Q) = 1 - \frac{2 \sqrt{\alpha \beta}}{\alpha + \beta}.$

The squared Hellinger distance between two Weibull distributions $\scriptstyle P\,\sim \,\rm{W}(\alpha,\beta)$ and $\scriptstyle Q\,\sim\,\rm{W}(\alpha,d)$ (where α is a common shape parameter and $\beta\, , d$ are the scale parameters respectively):

$H^2(P, Q) = 1 - \frac{2 (\beta d)^{\alpha/2}}{\beta^{\alpha} + d^{\alpha}}.$

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.