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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.
Applying this theorem to KL-divergence yields the Donsker–Varadhan representation. Attempting to apply this theorem to the general α {\displaystyle \alpha } -divergence with α ∈ ( − ∞ , 0 ) ∪ ( 0 , 1 ) {\displaystyle \alpha \in (-\infty ,0)\cup (0,1)} does not yield a closed-form solution.
The term "divergence" is in contrast to a distance (metric), since the symmetrized divergence does not satisfy the triangle inequality. [10] Numerous references to earlier uses of the symmetrized divergence and to other statistical distances are given in Kullback (1959 , pp. 6–7, §1.3 Divergence).
In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be between an individual sample point and a population or a wider sample of points.
The total variation distance is related to the Kullback–Leibler divergence by Pinsker’s inequality: (,) ().One also has the following inequality, due to Bretagnolle and Huber [2] (see also [3]), which has the advantage of providing a non-vacuous bound even when () >:
The only divergence for probabilities over a finite alphabet that is both an f-divergence and a Bregman divergence is the Kullback–Leibler divergence. [8] The squared Euclidean divergence is a Bregman divergence (corresponding to the function x 2 {\displaystyle x^{2}} ) but not an f -divergence.
Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection is the "closest" distribution to q of all the distributions in P. The I-projection is useful in setting up information geometry , notably because of the following inequality, valid when P is convex: [ 1 ]
In statistics, the Bhattacharyya distance is a quantity which represents a notion of similarity between two probability distributions. [1] It is closely related to the Bhattacharyya coefficient , which is a measure of the amount of overlap between two statistical samples or populations.