text: In applications,  typically represents the "true" distribution of data, observations, or a precisely calculated theoretical distribution, while  typically represents a theory, model, description, or approximation of . In order to find a distribution  that is closest to , we can minimize the KL divergence and compute an information projection.
  While it is a statistical distance, it is not a metric, the most familiar type of distance, but instead it is a divergence. While metrics are symmetric and generalize linear distance, satisfying the triangle inequality, divergences are asymmetric and generalize squared distance, in some cases satisfying a generalized Pythagorean theorem. In general  does not equal , and the asymmetry is an important part of the geometry. The infinitesimal form of relative entropy, specifically its Hessian, gives a metric tensor that equals the Fisher information metric; see . Relative entropy satisfies a generalized Pythagorean theorem for exponential families (geometrically interpreted as dually flat manifolds), and this allows one to minimize relative entropy by geometric means, for example by information projection and in maximum likelihood estimation.
  The relative entropy is the Bregman divergence generated by the negative entropy, but it is also of the form of an -divergence. For probabilities over a finite alphabet, it is unique in being a member of both of these classes of statistical divergences.