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Ali & Silvey (1966) referred generally to such a function as a "coefficient of divergence", and showed that many existing functions could be expressed as f-divergences, referring to Jeffreys' function as "Jeffreys' measure of divergence" (today "Jeffreys divergence"), and Kullback–Leibler's asymmetric function (in each direction) as "Kullback ...
In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence [1]), denoted (), is a type of statistical distance: a measure of how much a model probability distribution Q is different from a true probability distribution P.
In probability theory, an -divergence is a certain type of function (‖) that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence , Hellinger distance , and total variation distance , are special cases of f {\displaystyle f} -divergence.
In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences.
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.
Quantum Jensen–Shannon divergence for = (,) and two density matrices is a symmetric function, everywhere defined, bounded and equal to zero only if two density matrices are the same. It is a square of a metric for pure states , [ 13 ] and it was recently shown that this metric property holds for mixed states as well.
By Chentsov’s theorem, the Fisher information metric on statistical models is the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. [3] [4] It can also be understood to be the infinitesimal form of the relative entropy (i.e., the Kullback–Leibler divergence); specifically, it is the Hessian of
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