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The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by the Boltzmann constant).
In this context, either an information-theoretical measure, such as functional clusters (Gerald Edelman and Giulio Tononi's functional clustering model and dynamic core hypothesis (DCH) [47]) or effective information (Tononi's integrated information theory (IIT) of consciousness [48] [49] [50]), is defined (on the basis of a reentrant process ...
A misleading [1] information diagram showing additive and subtractive relationships among Shannon's basic quantities of information for correlated variables and .The area contained by both circles is the joint entropy (,).
The information content, also called the surprisal or self-information, of an event is a function that increases as the probability () of an event decreases. When p ( E ) {\displaystyle p(E)} is close to 1, the surprisal of the event is low, but if p ( E ) {\displaystyle p(E)} is close to 0, the surprisal of the event is high.
The continuous version of discrete joint entropy is called joint differential (or continuous) entropy. Let and be a continuous random variables with a joint probability density function (,). The differential joint entropy (,) is defined as [3]: 249
There is an analogy between Shannon's basic "measures" of the information content of random variables and a measure over sets. Namely the joint entropy, conditional entropy, and mutual information can be considered as the measure of a set union, set difference, and set intersection, respectively (Reza pp. 106–108).
Hence the use of the semicolon (or occasionally a colon or even a wedge ) to separate the principal arguments of the mutual information symbol. (No such distinction is necessary in the symbol for joint entropy, since the joint entropy of any number of random variables is the same as the entropy of their joint distribution.)
For a given probability space, the measurement of rarer events are intuitively more "surprising", and yield more information content, than more common values. Thus, self-information is a strictly decreasing monotonic function of the probability, or sometimes called an "antitonic" function.