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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 ...
In the simple version above, the signal and noise are fully uncorrelated, in which case + is the total power of the received signal and noise together. A generalization of the above equation for the case where the additive noise is not white (or that the / is not constant with frequency over the bandwidth) is obtained by treating the channel as many narrow, independent Gaussian ...
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:
Although, in both cases, mutual information expresses the number of bits of information common to the two sources in question, the analogy does not imply identical properties; for example, differential entropy may be negative. The differential analogies of entropy, joint entropy, conditional entropy, and mutual information are defined as follows:
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).
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.
In contrast to the conditional entropy for discrete random variables, the conditional differential entropy may be negative. As in the discrete case there is a chain rule for differential entropy: (|) = (,) [3]: 253
If the base of the logarithm is 2, then the unit of uncertainty is the shannon (more commonly known as bit). If it is the natural logarithm, then the unit is the nat. Hartley used a base-ten logarithm, and with this base, the unit of information is called the hartley (aka ban or dit) in his honor. It is also known as the Hartley entropy or max ...
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