![]() ![]() "On measures of information and entropy". 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.Ī. The other Rényi divergences satisfy the criteria of being positive and continuous being invariant under 1-to-1 co-ordinate transformations and of combining additively when A and X are independent, so that if p( A, X) = p( A) p( X), then 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( x| a) remains m( x| a), unchanged. The 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: : the log of the maximum ratio of the probabilities. : the log of the expected ratio of the probabilities : minus twice the logarithm of the Bhattacharyya coefficient Like the Kullback-Leibler divergence, the Rényi generalised divergences are always non-negative. 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: They also lead to a spectrum of indices of fractal dimension.Īs 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 entropies are important in ecology and statistics as indices of diversity. 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 min-entropy. As, the limit exists asĪnd this is called Min-entropy, because it is smallest value of H α. Where Y is a random variable independent of X but identically distributed to X. Sometimes Renyi entropy refers only to the case α = 2, In the limit that α approaches 1, it can be shown that H α converges to Which is the logarithm of the cardinality of X, sometimes called the Hartley entropy of X. ![]() Otherwise the entropies are weakly decreasing as a function of α. If the probabilities are all the same then all the Rényi entropies of the distribution are equal, with H α( X)=log n. The Rényi entropy of order α, where α 0, is defined as In 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. ![]()
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