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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 ...
Smoothmax of (−x, x) versus x for various parameter values. Very smooth for =0.5, and more sharp for =8. For large positive values of the parameter >, the following formulation is a smooth, differentiable approximation of the maximum function. For negative values of the parameter that are large in absolute value, it approximates the minimum.
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 ).
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 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.
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function.The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than exponential (e.g. sub-Gaussian).
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
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.