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In probability theory, integral probability metrics are types of distance functions between probability distributions, defined by how well a class of functions can distinguish the two distributions. Many important statistical distances are integral probability metrics, including the Wasserstein-1 distance and the total variation distance .
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 information geometry definition of divergence (the subject of this article) was initially referred to by alternative terms, including "quasi-distance" Amari (1982, p. 369) and "contrast function" Eguchi (1985), though "divergence" was used in Amari (1985) for the α-divergence, and has become standard for the general class. [1] [2]
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is (,) =, that is, ‖ ‖ = (,) = {(): =, =} = [], where the expectation is taken with respect to the probability measure on the space where (,) lives, and the infimum is taken over all such with marginals and , respectively.
In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space. It is named after Leonid Vaseršteĭn .
Energy distance is a statistical distance between probability distributions.If X and Y are independent random vectors in R d with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of
Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1). Then given a non-empty set S and a function F : S × S → D+ where we denote F ( p , q ) by F p , q for every ( p , q ) ∈ S × S , the ordered pair ( S , F ) is said to be a ...
In statistics, Bhattacharyya angle, also called statistical angle, is a measure of distance between two probability measures defined on a finite probability space. It is defined as It is defined as Δ ( p , q ) = arccos BC ( p , q ) {\displaystyle \Delta (p,q)=\arccos \operatorname {BC} (p,q)}