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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 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 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.
Total variation distance is half the absolute area between the two curves: Half the shaded area above. In probability theory, the total variation distance is a statistical distance between probability distributions, and is sometimes called the statistical distance, statistical difference or variational distance.
The only divergence on that is both a Bregman divergence and an f-divergence is the Kullback–Leibler divergence. [ 6 ] If n ≥ 3 {\displaystyle n\geq 3} , then any Bregman divergence on Γ n {\displaystyle \Gamma _{n}} that satisfies the data processing inequality must be the Kullback–Leibler divergence.
More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the ...
Larger values of the discrepancy function indicate a poor fit of the model to data. In general, the parameter estimates for a given model are chosen so as to make the discrepancy function for that model as small as possible. Analogous concepts in statistics are known as goodness of fit or statistical distance, and include deviance and divergence.
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.