This post was a warm-up for the next post: Rényi differential privacy. Information theory clarifies data discussions.Why Kullback-Leibler divergence is not a distance.Then for α not equal to 1,Īs with Lebesgue norms, you use varying values of the parameter to emphasize various features. This quantity is also known as the Kullback-Leibler divergence. If qk is not None, then compute the relative entropy D sum (pk log (pk / qk)). If only probabilities pk are given, the Shannon entropy is calculated as H -sum (pk log (pk)). Let p without a subscript be the vector of all the p i. Calculate the Shannon entropy/relative entropy of given distribution (s). In the limit as α goes to ∞, the Rényi entropy of X converges to the negative log of the probability of the most probable outcome. This case is also called collision entropy and is used in quantum information theory. Rnyi entropy is defined as: Halpha(p1,p2.,pn)1/(1-alpha)ln(sum(i1)npialpha), where alpha>0, alpha1. That is, when people speak of Rényi entropy without qualification, they often have in mind the case α = 2. When the order α is not specified, it’s implicit default value is 2. When α = 1 we get the more familiar Shannon entropy: Renyi’s information transfer between financial time series. It is simply log 2 n, the log of the number of values X takes on with positive probability. Rényi transfer entropy merely suggests a higher risk of the predicted variable as opposed to a situation with positive Rényi transfer entropy where the risk about the future returns of the predicted variable is reduced by knowledge of the present. It was introduced by Hungarian mathematician Alfrd Rnyi in the. Then the H 0 is known as the max-entropy, or Hartley entropy. Rnyi entropy is a measure of the uncertainty or randomness of a probability distribution. Max-entropy: α = 0Īssume all the probabilities p i are positive. And for each discrete random variable X, H α is a monotone non-decreasing function of α. All are additive for independent random variables. For example, let X be the random variable defined on [1, ∞) with densityĮach value of α gives a possible entropy measure. The definition of Rényi entropy can be extended to continuous random variables byīut unlike the discrete case, Rényi entropy can be negative for continuous random variables, and so Rényi entropy is typically only used for discrete variables. Rényi entropy of continuous random variable In the case α = 1 or ∞ this expression means the limit as α approaches 1 or ∞ respectively. If a discrete random variable X has n possible values, where the ith outcome has probability p i, then the Rényi entropy of order α is defined to beįor 0 ≤ α ≤ ∞. The most common way of measuring information is Shannon entropy, but there are others. Rényi entropy, developed by Hungarian mathematician Alfréd Rényi, generalizes Shannon entropy and includes other entropy measures as special cases.
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