Properties
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 as it can be defined as
Hellinger distances are used in the theory of sequential and asymptotic statistics.
Read more about this topic: Hellinger Distance
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“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)