![]() This post was a warm-up for the next post: Rényi differential privacy. We first define Renyi entropy and partial Renyi entropy to measure the indeterminacy of uncertain random variables and examine their mathematical properties. Information theory clarifies data discussions This paper proposes a new type of entropy called Renyi entropy as an extension of logarithm entropy in an uncertain random environment and applies it to portfolio selection.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. Let p without a subscript be the vector of all the p i. 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. A definition of the Renyi entropy-power itself appears in, which is essentially Definition 5 below. This case is also called collision entropy and is used in quantum information theory. 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: ![]() It is simply log 2 n, the log of the number of values X takes on with positive probability. Then the H 0 is known as the max-entropy, or Hartley entropy. Max-entropy: α = 0Īssume all the probabilities p i are positive. Renyi’s information transfer between financial time series. And for each discrete random variable X, H α is a monotone non-decreasing function of α. 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. ![]() 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 Recently we employed entropy dynamics, a statistical inference tool that facilitates quantifying posterior probabilities of likely particle positions. 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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