Jensen–Shannon divergence

Consider the set ${\displaystyle M_{+}^{1}(A)}$ 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 (JSD) ${\displaystyle M_{+}^{1}(A)\times M_{+}^{1}(A)\rightarrow [0,\infty {})}$ is a symmetrized and smoothed version of the Kullback–Leibler divergence ${\displaystyle D(P\parallel Q)}$. It is defined by

${\displaystyle {\rm {JSD}}(P\parallel Q)={\frac {1}{2}}D(P\parallel M)+{\frac {1}{2}}D(Q\parallel M)}$

where ${\displaystyle M={\frac {1}{2}}(P+Q)}$

A generalization of the Jensen–Shannon divergence using abstract means (like the geometric or harmonic means) instead of the arithmetic mean was recently proposed.[6] The geometric Jensen–Shannon divergence (or G-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:

${\displaystyle {\rm {JSD}}_{\pi _{1},\ldots ,\pi _{n}}(P_{1},P_{2},\ldots ,P_{n})=H\left(\sum _{i=1}^{n}\pi _{i}P_{i}\right)-\sum _{i=1}^{n}\pi _{i}H(P_{i})}$

where ${\displaystyle \pi _{1},\ldots ,\pi _{n}}$ are weights that are selected for the probability distributions ${\displaystyle P_{1},P_{2},\ldots ,P_{n}}$ and ${\displaystyle H(P)}$ is the Shannon entropy for distribution ${\displaystyle P}$. For the two-distribution case described above,

${\displaystyle P_{1}=P,P_{2}=Q,\pi _{1}=\pi _{2}={\frac {1}{2}}.\ }$

The Jensen–Shannon divergence is bounded by 1 for two probability distributions, given that one uses the base 2 logarithm.[7]

${\displaystyle 0\leq {\rm {JSD}}(P\parallel Q)\leq 1}$

With this normalization, it is a lower bound on the total variation distance between P and Q:

${\displaystyle {\rm {JSD}}(P\parallel Q)\leq {\frac {1}{2}}\|P-Q\|_{1}={\frac {1}{2}}\sum _{\omega \in \Omega }|P(\omega )-Q(\omega )|.}$

For log base e, or ln, which is commonly used in statistical thermodynamics, the upper bound is ln(2):

${\displaystyle 0\leq {\rm {JSD}}(P\parallel Q)\leq \ln(2)}$

A more general bound, the Jensen–Shannon divergence is bounded by ${\displaystyle \log _{2}(n)}$ for more than two probability distributions, given that one uses the base 2 logarithm.[7]

${\displaystyle 0\leq {\rm {JSD}}_{\pi _{1},\ldots ,\pi _{n}}(P_{1},P_{2},\ldots ,P_{n})\leq \log _{2}(n)}$

The Jensen–Shannon divergence is the mutual information between a random variable ${\displaystyle X}$ associated to a mixture distribution between ${\displaystyle P}$ and ${\displaystyle Q}$ and the binary indicator variable ${\displaystyle Z}$ that is used to switch between ${\displaystyle P}$ and ${\displaystyle Q}$ to produce the mixture. Let ${\displaystyle X}$ be some abstract function on the underlying set of events that discriminates well between events, and choose the value of ${\displaystyle X}$ according to ${\displaystyle P}$ if ${\displaystyle Z=0}$ and according to ${\displaystyle Q}$ if ${\displaystyle Z=1}$, where ${\displaystyle Z}$ is equiprobable. That is, we are choosing ${\displaystyle X}$ according to the probability measure ${\displaystyle M=(P+Q)/2}$, and its distribution is the mixture distribution. We compute

${\displaystyle {\begin{aligned}I(X;Z)&=H(X)-H(X|Z)\\&=-\sum M\log M+{\frac {1}{2}}\left[\sum P\log P+\sum Q\log Q\right]\\&=-\sum {\frac {P}{2}}\log M-\sum {\frac {Q}{2}}\log M+{\frac {1}{2}}\left[\sum P\log P+\sum Q\log Q\right]\\&={\frac {1}{2}}\sum P\left(\log P-\log M\right)+{\frac {1}{2}}\sum Q\left(\log Q-\log M\right)\\&={\rm {JSD}}(P\parallel Q)\end{aligned}}}$

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 ${\displaystyle H(Z)=1}$. 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 ${\displaystyle Z}$.

One can apply the same principle to a joint distribution and the product of its two marginal distribution (in analogy to Kullback–Leibler divergence and mutual information) 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.[8]

The generalization of probability distributions on density matrices allows to define quantum Jensen–Shannon divergence (QJSD).[9][10] It is defined for a set of density matrices ${\displaystyle (\rho _{1},\ldots ,\rho _{n})}$ and a probability distribution ${\displaystyle \pi =(\pi _{1},\ldots ,\pi _{n})}$ as

${\displaystyle {\rm {QJSD}}(\rho _{1},\ldots ,\rho _{n})=S\left(\sum _{i=1}^{n}\pi _{i}\rho _{i}\right)-\sum _{i=1}^{n}\pi _{i}S(\rho _{i})}$

where ${\displaystyle S(\rho )}$ is the von Neumann entropy of ${\displaystyle \rho }$. 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 ${\displaystyle (\rho _{1},\ldots ,\rho _{n})}$ under the prior distribution ${\displaystyle \pi }$ (see Holevo's theorem).[11] Quantum Jensen–Shannon divergence for ${\displaystyle \pi =\left({\frac {1}{2}},{\frac {1}{2}}\right)}$ 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[12], and it was recently shown that this metric property holds for mixed states as well.[13][14] The Bures metric is closely related to the quantum JS divergence; it is the quantum analog of the Fisher information metric.

Nielsen introduced the skew K-divergence:[15] ${\displaystyle K_{\alpha }(p||q)=\mathrm {KL} (p||(1-\alpha )p+\alpha q)=\int p(x)\log {\frac {p(x)}{(1-\alpha )p(x)+\alpha q(x)}}\mathrm {d} x.}$ It follows a one-parametric family of Jensen–Shannon divergences, called the ${\displaystyle \alpha }$-Jensen–Shannon divergences: ${\displaystyle \mathrm {JS} _{\alpha }(p,q)={\frac {1}{2}}\left(K_{\alpha }(p||q)+K_{\alpha }(q||p)\right)=\mathrm {JS} _{\alpha }(q,p),}$ which includes the Jensen–Shannon divergence (for ${\displaystyle \alpha ={\frac {1}{2}}}$) and the half of the Jeffreys divergence (for ${\displaystyle \alpha =1}$).

The Jensen–Shannon divergence has been applied in bioinformatics and genome comparison,[16][17] in protein surface comparison,[18] in the social sciences,[19] in the quantitative study of history,[20] and in machine learning.[21]

• Frank Nielsen (2010). "A family of statistical symmetric divergences based on Jensen's inequality". arXiv:1009.4004 [cs.CV].

• Ruby gem for calculating JS divergence
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• statcomp R library for calculating complexity measures including Jensen-Shannon Divergence