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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. Shannon's theory defines a data communication system composed of three elements: a source of data, a communication channel, and a receiver. The "fundamental problem ...
The Shannon entropy (in nats) is: = = = and if entropy is measured in units of per nat, then the entropy is given by: = which is the Boltzmann entropy formula, where is the Boltzmann constant, which may be interpreted as the thermodynamic entropy per nat.
Intuitively, the entropy H X of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known. The entropy of a source that emits a sequence of N symbols that are independent and identically distributed (iid) is N ⋅ H bits (per message of N symbols).
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.
The shannon also serves as a unit of the information entropy of an event, which is defined as the expected value of the information content of the event (i.e., the probability-weighted average of the information content of all potential events). Given a number of possible outcomes, unlike information content, the entropy has an upper bound ...
When measuring entropy using the natural logarithm (ln), the unit of information entropy is called a "nat", but when it is measured using the base-2 logarithm, the unit of information entropy is called a "shannon" (alternatively, "bit"). This is just a difference in units, much like the difference between inches and centimeters.
In information theory, the source coding theorem (Shannon 1948) [2] informally states that (MacKay 2003, pg. 81, [3] Cover 2006, Chapter 5 [4]): N i.i.d. random variables each with entropy H(X) can be compressed into more than N H(X) bits with negligible risk of information loss, as N → ∞; but conversely, if they are compressed into fewer than N H(X) bits it is virtually certain that ...
The Shannon information is closely related to entropy, which is the expected value of the self-information of a random variable, quantifying how surprising the random variable is "on average". This is the average amount of self-information an observer would expect to gain about a random variable when measuring it.