# Hellinger distance

A distance between probability measures, expressed in terms of the Hellinger integral. Suppose that on a measurable space $( \mathfrak X , {\mathcal B} )$ a family of probability measures $\{ {\mathsf P} _ \theta \}$, $\theta \in \Theta$, is given that are all absolutely continuous relative to some $\sigma$- finite measure $\mu$ on ${\mathcal B}$.

The Hellinger distance between two measures ${\mathsf P} _ {\theta _ {1} }$ and ${\mathsf P} _ {\theta _ {2} }$( $\theta _ {1} , \theta _ {2} \in \Theta$) is defined by the formula

$$r ( \theta _ {1} , \theta _ {2} ) = \ \{ 2 [ 1 - H ( \theta _ {1} , \theta _ {2} )] \} ^ {1/2\ } =$$

$$= \ \left \{ \int\limits _ { \mathfrak X } \left [ \sqrt { \frac{d {\mathsf P} _ {\theta _ {1} } }{d \mu } } - \sqrt { \frac{d {\mathsf P} _ {\theta _ {2} } }{d \mu } } \right ] ^ {2} d \mu \right \} ^ {1/2} ,$$

where

$$H ( \theta _ {1} , \theta _ {2} ) = \ \int\limits _ { \mathfrak X } \sqrt { \frac{d {\mathsf P} _ {\theta _ {1} } }{d \mu } } \sqrt { \frac{d {\mathsf P} _ {\theta _ {2} } }{d \mu } } d \mu$$

is the Hellinger integral. The Hellinger distance does not depend on the choice of the measure $\mu$ and has the following properties:

1) $0 \leq r ( \theta _ {1} , \theta _ {2} ) \leq \sqrt 2$;

2) $r ( \theta _ {1} , \theta _ {2} ) = \sqrt 2$ if and only if the measures ${\mathsf P} _ {\theta _ {1} }$ and ${\mathsf P} _ {\theta _ {2} }$ are mutually singular;

3) $r ( \theta _ {1} , \theta _ {2} ) = 0$ if and only if ${\mathsf P} _ {\theta _ {1} } = {\mathsf P} _ {\theta _ {2} }$.

Let

$$\| {\mathsf P} _ {\theta _ {1} } - {\mathsf P} _ {\theta _ {2} } \| = \ \sup _ {B \in {\mathcal B} } \ | {\mathsf P} _ {\theta _ {1} } ( B) - {\mathsf P} _ {\theta _ {2} } ( B) | =$$

$$= \ { \frac{1}{2} } \int\limits _ { \mathfrak X } \left | \frac{d {\mathsf P} _ {\theta _ {1} } }{d \mu } - \frac{d {\mathsf P} _ {\theta _ {2} } }{d \mu } \right | d \mu$$

be the distance in variation between the measures ${\mathsf P} _ {\theta _ {1} }$ and ${\mathsf P} _ {\theta _ {2} }$. Then

$${ \frac{1}{2} } r ^ {2} ( \theta _ {1} , \theta _ {2} ) \leq \ \| {\mathsf P} _ {\theta _ {1} } - {\mathsf P} _ {\theta _ {2} } \| \leq \ r ( \theta _ {1} , \theta _ {2} ).$$

#### References

 [1] H.H. Kuo, "Gaussian measures on Banach spaces" , Springer (1975) [2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) [3] I.A. Ibragimov, R.Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian) [4] V.M. Zolotarev, "Properties and relations of certain types of metrics" Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk. USSR , 87 (1979) pp. 18–35; 206–212 (In Russian) (English summary)
How to Cite This Entry:
Hellinger distance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hellinger_distance&oldid=47206
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article