# Hellinger distance

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A distance between probability measures, expressed in terms of the Hellinger integral. Suppose that on a measurable space a family of probability measures , , is given that are all absolutely continuous relative to some -finite measure on .

The Hellinger distance between two measures and ( ) is defined by the formula  where is the Hellinger integral. The Hellinger distance does not depend on the choice of the measure and has the following properties:

1) ;

2) if and only if the measures and are mutually singular;

3) if and only if .

Let  be the distance in variation between the measures and . Then How to Cite This Entry:
Hellinger distance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hellinger_distance&oldid=16453
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article