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The entropy () thus sets a minimum value for the cross-entropy (,), the expected number of bits required when using a code based on Q rather than P; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must be transmitted to identify a value x drawn from X, if a code is used corresponding to 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 ).
When NMF is obtained by minimizing the Kullback–Leibler divergence, it is in fact equivalent to another instance of multinomial PCA, probabilistic latent semantic analysis, [45] trained by maximum likelihood estimation. That method is commonly used for analyzing and clustering textual data and is also related to the latent class model.
The total correlation is also the Kullback–Leibler divergence between the actual distribution (,, …,) and its maximum entropy product approximation () (). Total correlation quantifies the amount of dependence among a group of variables.
Truncated normals with fixed support form an exponential family. Nielsen [3] reported closed-form formula for calculating the Kullback-Leibler divergence and the Bhattacharyya distance between two truncated normal distributions with the support of the first distribution nested into the support of the second distribution.
Now we can formally define the conditional probability measure given the value of one (or, via the product topology, more) of the random variables. Let M {\displaystyle M} be a measurable subset of Ω , {\displaystyle \Omega ,} (i.e. M ∈ F , {\displaystyle M\in {\mathcal {F}},} ) and let x ∈ s u p p X . {\displaystyle x\in \mathrm {supp} \,X.}
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.
However, in the context of decision trees, the term is sometimes used synonymously with mutual information, which is the conditional expected value of the Kullback–Leibler divergence of the univariate probability distribution of one variable from the conditional distribution of this variable given the other one.