The value Îą = 1, which gives the Shannon entropy and the KullbackâLeibler divergence, is special because it is only at Îą=1 that the chain rule of conditional probability holds exactly: Some special cases: : minus the log probability under Q that p i>0 : minus twice the logarithm of the Bhattacharyya coefficient ( Nielsen & Boltz (2009)) : the Kullback-Leibler divergence : the log of the expected ratio of the probabilities : the log of the maximum ratio of the probabilities.įor any fixed distributions P and Q, RĂŠnyi divergence is nondecreasing as a function of its order Îą, and it is continuous on the set of Îą for which it is finite. Like the Kullback-Leibler divergence, the RĂŠnyi divergences are The RĂŠnyi divergence of order Îą, where Îą > 0, of a distribution P from a distribution Q is defined to be: RĂŠnyi divergence Īs well as the absolute RĂŠnyi entropies, RĂŠnyi also defined a spectrum of divergence measures generalising the KullbackâLeibler divergence. On the other hand, the Shannon entropy can be arbitrarily high for a random variable that has a given min-entropy. , įor values of, inequalities in the other direction also hold. In particular cases inequalities can be proven also by Jensen's inequality. Which is proportional to KullbackâLeibler divergence (which is always non-negative), where. That is non-increasing in, which can be proven by differentiation, as Inequalities between different values of Îą The min-entropy has important applications for randomness extractors in theoretical computer science: Extractors are able to extract randomness from random sources that have a large min-entropy merely having a large Shannon entropy does not suffice for this task. In particular, the min-entropy is never larger Sense, it is the strongest way to measure the information content of aĭiscrete random variable. Smallest entropy measure in the family of RĂŠnyi entropies. The name min-entropy stems from the fact that it is the In the limit as, the RĂŠnyi entropy converges to the min-entropy :Įquivalently, the min-entropy is the largest real number such that all events occur with probability at most. The RĂŠnyi entropy for any is Schur concave. Here, the discrete probability distribution is interpreted as a vector in with and. Īpplications often exploit the following relation between the RĂŠnyi entropy and the p-norm of the vector of probabilities. In general, for all discrete random variables, is a non-increasing function in. If the probabilities are for all, then all the RĂŠnyi entropies of the distribution are equal. Here, is a discrete random variable with possible outcomes and corresponding probabilities for, and the logarithm is base 2. The RĂŠnyi entropy of order, where and, is defined as. 3 Inequalities between different values of Îą.
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