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Generally, information entropy is the average amount of information conveyed by an event, when considering all possible outcomes. The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication", [2] [3] and is also referred to as Shannon entropy.
Despite the foregoing, there is a difference between the two quantities. The information entropy Η can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p i occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p i specifically.
Shannon originally wrote down the following formula for the entropy of a continuous distribution, known as differential entropy: = ().Unlike Shannon's formula for the discrete entropy, however, this is not the result of any derivation (Shannon simply replaced the summation symbol in the discrete version with an integral), and it lacks many of the properties that make the discrete entropy a ...
The mathematical theory of information is based on probability theory and statistics, and measures information with several quantities of information. The choice of logarithmic base in the following formulae determines the unit of information entropy that is used.
Many of the concepts in information theory have separate definitions and formulas for continuous and discrete cases. For example, entropy is usually defined for discrete random variables, whereas for continuous random variables the related concept of differential entropy, written (), is used (see Cover and Thomas, 2006, chapter 8).
Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the idea of (Shannon) entropy (a measure of average surprisal) of a random variable, to continuous probability distributions. Unfortunately, Shannon did not derive this formula, and rather just ...
Entropy of a Bernoulli trial (in shannons) as a function of binary outcome probability, called the binary entropy function. In information theory, the binary entropy function, denoted or (), is defined as the entropy of a Bernoulli process (i.i.d. binary variable) with probability of one of two values, and is given by the formula:
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.