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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).
Where the logarithm is taken in basis 2 to obtain the mutual information in bits. But this is precisely the relative entropy between p(x, y) and p(x)p(y). In other words, if we assume the two variables x and y to be uncorrelated, mutual information is the discrepancy in uncertainty resulting from this (possibly erroneous) assumption.
Two bits of entropy: In the case of two fair coin tosses, the information entropy in bits is the base-2 logarithm of the number of possible outcomes — with two coins there are four possible outcomes, and two bits of entropy. Generally, information entropy is the average amount of information conveyed by an event, when considering all ...
The violet is the mutual information (;). 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
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 expected value of the information gain is the mutual information (;) of and – i.e. the reduction in the entropy of achieved by learning the state of the random variable . In machine learning, this concept can be used to define a preferred sequence of attributes to investigate to most rapidly narrow down the state of X .
Other important information theoretic quantities include the Rényi entropy and the Tsallis entropy (generalizations of the concept of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information.
An information diagram is a type of Venn diagram used in information theory to illustrate relationships among Shannon's basic measures of information: entropy, joint entropy, conditional entropy and mutual information. [1] [2] Information