In statistics and information geometry, divergence or a contrast function is a function which establishes the "distance" of one probability distribution to the other on a statistical manifold. The divergence is a weaker notion than that of the distance, in particular the divergence need not be symmetric (that is, in general the divergence from p to q is not equal to the divergence from q to p), and need not satisfy the triangle inequality.
Suppose S is a space of all probability distributions with common support. Then a divergence on S is a function D(· || ·): S×S → R satisfying
- D(p || q) ≥ 0 for all p, q ∈ S,
- D(p || q) = 0 if and only if p = q,
The dual divergence D* is defined as
Many properties of divergences can be derived if we restrict S to be a statistical manifold, meaning that it can be parametrized with a finite-dimensional coordinate system θ, so that for a distribution p ∈ S we can write p = p(θ).
For a pair of points p, q ∈ S with coordinates θp and θq, denote the partial derivatives of D(p || q) as
By definition, the function D(p || q) is minimized at p = q, and therefore
and the dual to this connection ∇* is generated by the dual divergence D*.
Thus, a divergence D(· || ·) generates on a statistical manifold a unique dualistic structure (g(D), ∇(D), ∇(D*)). The converse is also true: every torsion-free dualistic structure on a statistical manifold is induced from some globally defined divergence function (which however need not be unique).
For example, when D is an f-divergence for some function ƒ(·), then it generates the metric g(Df) = c·g and the connection ∇(Df) = ∇(α), where g is the canonical Fisher information metric, ∇(α) is the α-connection, c = ƒ′′(1), and α = 3 + 2ƒ′′′(1)/ƒ′′(1).
The largest and most frequently used class of divergences form the so-called f-divergences, however other types of divergence functions are also encountered in the literature.
This family of divergences are generated through functions f(u), convex on u > 0 and such that f(1) = 0. Then an f-divergence is defined as
|squared Hellinger distance:|
- Amari, Shun-ichi; Nagaoka, Hiroshi (2000). Methods of information geometry. Oxford University Press. ISBN 0-8218-0531-2.
- Eguchi, Shinto (1985). "A differential geometric approach to statistical inference on the basis of contrast functionals". Hiroshima mathematical journal. 15 (2): 341–391.
- Eguchi, Shinto (1992). "Geometry of minimum contrast". Hiroshima mathematical journal. 22 (3): 631–647.
- Matumoto, Takao (1993). "Any statistical manifold has a contrast function — on the C³-functions taking the minimum at the diagonal of the product manifold". Hiroshima mathematical journal. 23 (2): 327–332.