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Concept
Rényi entropy
Summary
In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions.
The Rényi entropy is important in ecology and statistics as index of diversity. The Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement. In the Heisenberg XY spin chain model, the Rényi entropy as a function of α can be calculated explicitly because it is an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in the context of randomness extractors.
The Rényi entropy of order , where and , is defined as
It is further defined at as
Here, is a discrete random variable with possible outcomes in the set and corresponding probabilities for . The resulting unit of information is determined by the base of the logarithm, e.g. shannon for base 2, or nat for base e.
If the probabilities are for all , then all the Rényi entropies of the distribution are equal: .
In general, for all discrete random variables , is a non-increasing function in .
Applications often exploit the following relation between the Rényi entropy and the p-norm of the vector of probabilities:
Here, the discrete probability distribution is interpreted as a vector in with and .
The Rényi entropy for any is Schur concave.
As approaches zero, the Rényi entropy increasingly weighs all events with nonzero probability more equally, regardless of their probabilities. In the limit for , the Rényi entropy is just the logarithm of the size of the support of X. The limit for is the Shannon entropy. As approaches infinity, the Rényi entropy is increasingly determined by the events of highest probability.
Official source
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