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
is the Hellinger integral. The Hellinger distance does not depend on the choice of the measure and has the following properties:
2) if and only if the measures and are mutually singular;
3) if and only if .
be the distance in variation between the measures and . Then
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Hellinger distance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hellinger_distance&oldid=16453