In probability theory and statistics, the Jensen–Shannon divergence is a method of measuring the similarity between two probability distributions. It is also known as information radius or total divergence to the average. It is based on the Kullback–Leibler divergence, with some notable differences, including that it is symmetric and it always has a finite value. The square root of the Jensen–Shannon divergence is a metric often referred to as Jensen-Shannon distance.
Definition
Consider the set of probability distributions where A is a set provided with some σ-algebra of measurable subsets. In particular we can take A to be a finite or countable set with all subsets being measurable. The Jensen–Shannon divergence is a symmetrized and smoothed version of the Kullback–Leibler divergence. It is defined by where A generalization of the Jensen–Shannon divergence using abstract means instead of the arithmetic mean was recently proposed. The geometric Jensen–Shannon divergence yields a closed-form formula for divergence between two Gaussian distributions by taking the geometric mean. A more general definition, allowing for the comparison of more than two probability distributions, is: where are weights that are selected for the probability distributions and is the Shannon entropy for distribution. For the two-distribution case described above,
Bounds
The Jensen–Shannon divergence is bounded by 1 for two probability distributions, given that one uses the base 2 logarithm. With this normalization, it is a lower bound on the total variation distance between P and Q: For log base e, or ln, which is commonly used in statistical thermodynamics, the upper bound is ln: A more general bound, the Jensen–Shannon divergence is bounded by for more than two probability distributions, given that one uses the base 2 logarithm.
The Jensen–Shannon divergence is the mutual information between a random variable associated to a mixture distribution between and and the binary indicator variable that is used to switch between and to produce the mixture. Let be some abstract function on the underlying set of events that discriminates well between events, and choose the value of according to if and according to if, where is equiprobable. That is, we are choosing according to the probability measure , and its distribution is the mixture distribution. We compute It follows from the above result that the Jensen–Shannon divergence is bounded by 0 and 1 because mutual information is non-negative and bounded by. The JSD is not always bounded by 0 and 1: the upper limit of 1 arises here because we are considering the specific case involving the binary variable. One can apply the same principle to a joint distribution and the product of its two marginal distribution and to measure how reliably one can decide if a given response comes from the joint distribution or the product distribution—subject to the assumption that these are the only two possibilities.
Quantum Jensen–Shannon divergence
The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence. It is defined for a set of density matrices and a probability distribution as where is the von Neumann entropy of. This quantity was introduced in quantum information theory, where it is called the Holevo information: it gives the upper bound for amount of classical information encoded by the quantum states under the prior distribution . 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, and it was recently shown that this metric property holds for mixed states as well. The Bures metric is closely related to the quantum JS divergence; it is the quantum analog of the Fisher information metric.
Generalization
Nielsen introduced the skew K-divergence: It follows a one-parametric family of Jensen–Shannon divergences, called the -Jensen–Shannon divergences: which includes the Jensen–Shannon divergence and the half of the Jeffreys divergence.
