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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.
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 () >:
Notably, except for total variation distance, all others are special cases of -divergence, or linear sums of -divergences. For each f-divergence D f {\displaystyle D_{f}} , its generating function is not uniquely defined, but only up to c ⋅ ( t − 1 ) {\displaystyle c\cdot (t-1)} , where c {\displaystyle c} is any real constant.
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).
Bhattacharyya distance (despite its name it is not a distance, as it violates the triangle inequality) f-divergence : generalizes several distances and divergences Discriminability index , specifically the Bayes discriminability index , is a positive-definite symmetric measure of the overlap of two distributions.
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.
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.
Hellinger distance; K. Kullback–Leibler divergence; T. Total variation distance of probability measures This page was last edited on 3 April 2023, at 01:17 (UTC). ...