collision entropy is related to the index of coincidence. Min-entropy In the limit as , the Rényi entropy converges to the min-entropy : Equivalently, the min-entropy is the largest real number such that all events occur with probability at most . The name min-entropy stems from the fact that it is the smallest entropy measure in the family of Rényi entropies. In this sense, it is the strongest way to measure the information content of a discrete random variable. In particular, the min-entropy is never larger than the Shannon entropy. 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. Inequalities for different orders α That is non-increasing in for any given distribution of probabilities , which can be proven by differentiation, as which is proportional to Kullback–Leibler divergence (which is always non-negative), where . In particular cases inequalities can be proven also by Jensen's inequality: For values of , inequalities in the other direction also hold. In particular, we have On the other hand, the Shannon entropy can be arbitrarily high for a random variable that has a given min-entropy. An example of this is given by the sequence of random variables for such that and since but . Rényi divergence As well as the absolute Rényi entropies, Rényi also defined a spectrum of divergence measures generalising the Kullback–Leibler divergence. The Rényi divergence of order or alpha-divergence of a distribution from a distribution is defined to be when and . We can define the Rényi divergence for the special values by taking a limit, and in particular the limit gives the Kullback–Leibler divergence. Some special cases: : minus the log probability under that ; : minus twice the logarithm of the Bhattacharyya coefficient; () : the Kullback–Leibler divergence; : the log of the expected ratio of the probabilities; : the log of the maximum ratio of the probabilities. The Rényi divergence is indeed a divergence, meaning simply that is greater than or equal to zero, and zero only when . For any fixed distributions and , the Rényi divergence is nondecreasing as a function of its order , and it is continuous on the set of for which it is finite, or for the sake of brevity, the information of order