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Numerous references to earlier uses of the symmetrized divergence and to other statistical distances are given in Kullback (1959, pp. 6–7, §1.3 Divergence). The asymmetric "directed divergence" has come to be known as the Kullback–Leibler divergence, while the symmetrized "divergence" is now referred to as the Jeffreys divergence.
For the classical Kullback–Leibler divergence, it can be shown that (‖) = , and the equality holds if and only if P = Q. Colloquially, this means that the uncertainty calculated using erroneous assumptions is always greater than the real amount of uncertainty.
where is the Kullback–Leibler divergence, and is the outer product distribution which assigns probability () to each (,).. Notice, as per property of the Kullback–Leibler divergence, that (;) is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when and are independent (and hence observing tells you nothing about ).
Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection is the "closest" distribution to q of all the distributions in P. The I-projection is useful in setting up information geometry , notably because of the following inequality, valid when P is convex: [ 1 ]
Since =, you can obtain previous formula. Also note that p i i = 0 {\displaystyle p_{ii}=0} and ∑ i , j p i j = 1 {\displaystyle \sum _{i,j}p_{ij}=1} . The bandwidth of the Gaussian kernels σ i {\displaystyle \sigma _{i}} is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection ...
In information theory, Pinsker's inequality, named after its inventor Mark Semenovich Pinsker, is an inequality that bounds the total variation distance (or statistical distance) in terms of the Kullback–Leibler divergence. The inequality is tight up to constant factors. [1]
As well as the absolute Rényi entropies, Rényi also defined a spectrum of divergence measures generalising the Kullback–Leibler divergence. [ 13 ] The Rényi divergence of order α {\displaystyle \alpha } or alpha-divergence of a distribution P from a distribution Q is defined to be
Alternatively, the metric can be obtained as the second derivative of the relative entropy or Kullback–Leibler divergence. [5] To obtain this, one considers two probability distributions P ( θ ) {\displaystyle P(\theta )} and P ( θ 0 ) {\displaystyle P(\theta _{0})} , which are infinitesimally close to one another, so that