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Rényi entropyIn information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. It is named after Alfréd Rényi. The Rényi entropy of order α, where α 0, is defined as where p_{i} are the probabilities of {x_{1}, x_{2} ... x_{n}}. If the probabilities are all the same then all the Rényi entropies of the distribution are equal, with H_{α}(X)=log n. Otherwise the entropies are weakly decreasing as a function of α. Some particular cases: which is the logarithm of the cardinality of X, sometimes called the Hartley entropy of X. In the limit that α approaches 1, it can be shown that H_{α} converges to which is the Shannon entropy. Sometimes Renyi entropy refers only to the case α = 2, where Y is a random variable independent of X but identically distributed to X. As , the limit exists as and this is called Minentropy, because it is smallest value of H_{α}. These two latter cases are related by , while on the other hand Shannon entropy can be arbitrarily high for a random variable X with fixed minentropy. The Rényi entropies are important in ecology and statistics as indices of diversity. They also lead to a spectrum of indices of fractal dimension. Additional recommended knowledge
Rényi relative informationsAs well as the absolute Rényi entropies, Rényi also defined a spectrum of generalised relative information gains (the negative of relative entropies), generalising the Kullback–Leibler divergence. The Rényi generalised divergence of order α, where α > 0, of an approximate distribution or a prior distribution Q(x) from a "true" distribution or an updated distribution P(x) is defined to be: Like the KullbackLeibler divergence, the Rényi generalised divergences are always nonnegative. Some special cases:
Why α = 1 is specialThe value α = 1, which gives the Shannon entropy and the Kullback–Leibler divergence, is special because it is only when α=1 that one can separate out variables A and X from a joint probability distribution, and write: for the absolute entropies, and for the relative entropies. The latter in particular means that if we seek a distribution p(x,a) which minimises the divergence of some underlying prior measure m(x,a), and we acquire new information which only affects the distribution of a, then the distribution of p(xa) remains m(xa), unchanged. The other Rényi divergences satisfy the criteria of being positive and continuous; being invariant under 1to1 coordinate transformations; and of combining additively when A and X are independent, so that if p(A,X) = p(A)p(X), then and The stronger properties of the α = 1 quantities, which allow the definition of the conditional informations and mutual informations which are so important in communication theory, may be very important in other applications, or entirely unimportant, depending on those applications' requirements. ReferencesA. Rényi (1961). "On measures of information and entropy". Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 1960: 547561. See also


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Rényi_entropy". A list of authors is available in Wikipedia. 