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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 ).
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.
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.
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
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, an -divergence is a certain type of function (‖) that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence , Hellinger distance , and total variation distance , are special cases of f {\displaystyle f} -divergence.
In our coin-tossing the value of the "rate function" for mean value equal to 1/2 is zero. In this way one can see the "rate function" as the negative of the "entropy". There is a relation between the "rate function" in large deviations theory and the Kullback–Leibler divergence , the connection is established by Sanov's theorem (see Sanov ...