In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya 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 in 1909.
[1][2] It is sometimes called the Jeffreys distance.
[3][4] To define the Hellinger distance in terms of measure theory, let
denote two probability measures on a measure space
that are absolutely continuous with respect to an auxiliary measure
The square of the Hellinger distance between
This definition does not depend on
, i.e. the Hellinger distance between P and Q does not change if
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 To define the Hellinger distance in terms of elementary probability theory, we take λ to be the 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 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 equals 1.
The Hellinger distance H(P, Q) satisfies the property (derivable from the Cauchy–Schwarz inequality) For two discrete probability distributions
, their Hellinger distance is defined as which is directly related to the Euclidean norm of the difference of the square root vectors, i.e. Also,
[citation needed] The Hellinger distance forms a bounded metric on the space of probability distributions over a given probability space.
The maximum distance 1 is achieved when P assigns probability zero to every set to which Q assigns a positive probability, and vice versa.
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.
[5][6] The squared Hellinger distance between two normal distributions
is: The squared Hellinger distance between two multivariate normal distributions
is [7] The squared Hellinger distance between two exponential distributions
is: The squared Hellinger distance between two Weibull distributions
is a common shape parameter and
are the scale parameters respectively): The squared Hellinger distance between two Poisson distributions with rate parameters
, is: The squared Hellinger distance between two beta distributions
The squared Hellinger distance between two gamma distributions
and the total variation distance (or statistical distance)
are related as follows:[8] The constants in this inequality may change depending on which renormalization you choose (