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In information geometry, a divergence is a kind of statistical distance: a binary function which establishes the separation from one probability distribution to another on a statistical manifold. The simplest divergence is squared Euclidean distance (SED), and divergences can be viewed as generalizations of SED.
The Dirac comb of period 2 π, although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrapped Dirac delta function. It represents a discrete probability distribution concentrated at 2 π n — a degenerate distribution — but the notation treats it as if it were a continuous distribution.
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).
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.
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.
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.
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.
As the number of discrete events increases, the function begins to resemble a normal distribution. Comparison of probability density functions, () for the sum of fair 6-sided dice to show their convergence to a normal distribution with increasing , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles ...