Annals of Mathematical Statistics 22(1):79–86, March 1951. It should be noted that Kullback and Leibler themselves actually defined the divergence as: In quantum information science it is used as a measure of entanglement in a state. In Bayesian experimental design a design which is optimised to maximise the KL divergence between the prior and the posterior is said to be Bayes d-optimal. The KL divergence is also the gain in Shannon information involved in going from the prior to the posterior. In Bayesian statistics the KL divergence can be used as a measure of the "distance" between the prior distribution and the posterior distribution. In coding theory, the KL divergence can be interpreted as the needed extra message-length per datum for sending messages distributed as q, if the messages are encoded using a code that is optimal for distribution p. As the cross-entropy is always greater than or equal to the entropy, this shows that the Kullback-Leibler divergence is nonnegative, and furthermore KL( p, q) is zero iff p = q, a result known as Gibbs' inequality. KL( p, q) = − ∑ x p( x)log 2 q( x) + ∑ x p( x)log 2 p( x) = H( p, q) − H( p)ĭenoting by H( p, q) the cross entropy of p and q, and by H( p) the entropy of p. 2 3 A simple interpretation of the KL divergence of P from Q is the ex. The logarithms in these formulae are conventionally taken to base 2, so that the quantity can be interpreted in units of bits the other important properties of the KL divergence hold irrespective of log base. In mathematical statistics, the KullbackLeibler divergence (also called relative entropy and I-divergence 1 ), denoted, is a type of statistical distance: a measure of how one probability distribution P is different from a second, reference probability distribution Q. $$\mathrm \ dx \!$$įor distributions of a continuous random variable. The Kullback-Leibler divergence between two probability distributions p and q is defined as One might be tempted to call it a " distance metric", but this would also be a misnomer as the Kullback-Leibler divergence is not symmetric and does not satisfy the triangle inequality. The term "divergence" is a misnomer it is not the same as divergence in calculus. It is proposed by Kullback and Leibler 58. It is named after Solomon Kullback and Richard Leibler. Relative entropy (KullbackLeibler divergence) is a basic conception in probability theory and information theory. That is, the relative entropy is jointly semicontinuous.In probability theory and information theory, the Kullback-Leibler divergence, or relative entropy, is a quantity which measures the difference between two probability distributions. Since the right hand side is the supremum of continuous functions in both $\mu$ and $\omega$ we can deduce that if $\mu_n \rightharpoonup \mu$ and $\omega_n \rightharpoonup \omega$ then This is reflected by non-finite quantum relative entropy for orthogonal quantum states. The KL divergence, which is closely related to relative entropy, informa-tion divergence, and information for discrimination, is a non-symmetric mea-sure of the dierence between two probability distributions p(x) and q(x). Being orthogonal represents the most different quantum states can be. The resulting method differs signif- icantly from previous policy gradient. Let $M$ be a closed manifold and $\mathcal. Informally, the quantum relative entropy is a measure of our ability to distinguish two quantum states where larger values indicate states that are more different. path of reasoning and suggest the Relative Entropy Policy. Following is the definition we will use in this question. There are several different definitions of relative entropy, and some of them are not equivalent.
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