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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 H ( X , Y ) {\displaystyle \mathrm {H} (X,Y)} .
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 "rate of self-information" can also be defined for a particular sequence of messages or symbols generated by a given stochastic process: this will always be equal to the entropy rate in the case of a stationary process.) Other quantities of information are also used to compare or relate different sources of information.
The landmark event establishing the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948.
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).
Therefore, the choice of the base b determines the unit used to measure information. In particular, if b is a positive integer, then the unit is the amount of information that can be stored in a system with b possible states. When b is 2, the unit is the shannon, equal to the information content of one "bit".
Taking into account these properties, the self-information associated with outcome with probability is defined as: = ( ()) = ( ()) The smaller the probability of event ω n {\displaystyle \omega _{n}} , the larger the quantity of self-information associated with the message that the event indeed occurred.
If we knew f, then we could find the information lost from using g 1 to represent f by calculating the Kullback–Leibler divergence, D KL (f ‖ g 1); similarly, the information lost from using g 2 to represent f could be found by calculating D KL (f ‖ g 2). We would then, generally, choose the candidate model that minimized the information ...