Search results
Results from the WOW.Com Content Network
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 much a model probability distribution Q is different from a true probability distribution P.
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 ]
Here, [] is the kernel embedding of the proposed density and is an entropy-like quantity (e.g. Entropy, KL divergence, Bregman divergence). The distribution which solves this optimization may be interpreted as a compromise between fitting the empirical kernel means of the samples well, while still allocating a substantial portion of the ...
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 ).
The only divergence for probabilities over a finite alphabet that is both an f-divergence and a Bregman divergence is the Kullback–Leibler divergence. [8] The squared Euclidean divergence is a Bregman divergence (corresponding to the function x 2 {\displaystyle x^{2}} ) but not an f -divergence.
In probability theory, particularly information theory, the conditional mutual information [1] [2] is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third.
The Rényi divergence is indeed a divergence, meaning simply that (‖) is greater than or equal to zero, and zero only when P = Q. For any fixed distributions P and Q , the Rényi divergence is nondecreasing as a function of its order α , and it is continuous on the set of α for which it is finite, [ 13 ] or for the sake of brevity, the ...
Note that the expression of Pinsker inequality depends on what basis of logarithm is used in the definition of KL-divergence. () = .Given the above comments, there is an alternative statement of Pinsker's inequality in some literature that relates information divergence to variation distance: