Kullback–Leibler divergence
Mathematical statistics distance measure / From Wikipedia, the free encyclopedia
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 one probability distribution P is different from a second, reference probability distribution Q.[2][3] Mathematically, it is defined as
A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model instead of P when the actual distribution is P. While it is a measure of how different two distributions are, and in some sense is thus a "distance", it is not actually a metric, which is the most familiar and formal type of distance. In particular, it is not symmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence,[4] a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances).[5]
Relative entropy is always a non-negative real number, with value 0 if and only if the two distributions in question are identical. It has diverse applications, both theoretical, such as characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series, and information gain when comparing statistical models of inference; and practical, such as applied statistics, fluid mechanics, neuroscience, bioinformatics, and machine learning.