Hellinger distance

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.
Definition.
Measure theory.
To define the Hellinger distance in terms of measure theory, let "P" and "Q" denote two probability measures that are absolutely continuous with respect to a third probability measure λ. The square of the Hellinger distance between "P" and "Q" is defined as the quantity
Here, "dP" / "dλ" and "dQ" / "d"λ are the Radon–Nikodym derivatives of "P" and "Q" respectively. This definition does not depend on λ, so the Hellinger distance between "P" and "Q" does not change if λ 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
Probability theory using Lebesgue measure.
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" / "d"λ 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)
Discrete distributions.
For two discrete probability distributions formula_5 and formula_6,
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, formula_9
Properties.
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.
Sometimes the factor formula_10 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 formula_11 as it can be defined as
Hellinger distances are used in the theory of sequential and asymptotic statistics.
The squared Hellinger distance between two normal distributions formula_13 and formula_14 is:
The squared Hellinger distance between two multivariate normal distributions formula_16 and formula_17 is 
The squared Hellinger distance between two exponential distributions formula_19 and formula_20 is:
The squared Hellinger distance between two Weibull distributions formula_22 and formula_23 (where formula_24 is a common shape parameter and formula_25 are the scale parameters respectively):
The squared Hellinger distance between two Poisson distributions with rate parameters formula_27 and formula_28, so that formula_29 and formula_30, is:
The squared Hellinger distance between two Beta distributions formula_32 and formula_33 is:
where formula_35 is the Beta function.
Connection with total variation distance.
The Hellinger distance formula_36 and the total variation distance (or statistical distance) formula_37 are related as follows:
These inequalities follow immediately from the inequalities between the 1-norm and the 2-norm.