Information entropy

Claude E. Shannon defined a measure of entropy (H = - Σ p_i log₂ p_i) that, when applied to an information source, could determine the minimum channel capacity required to reliably transmit the source as encoded binary digits. Shannon's entropy measure came to be taken as a measure of the uncertainty about the realization of a random variable. It thus served as a proxy capturing the concept of information contained in a message as opposed to the portion of the message that is strictly determined (hence predictable) by inherent structures. For example, redundancy in language structure or statistical properties relating to the occurrence frequencies of letter or word pairs, triplets etc. See Markov chains.

Shannon's definition of Entropy is closely related to thermodynamic entropy as defined by physicists and many chemists. Boltzmann and Gibbs did considerable work on statistical thermodynamics, which became the inspiration for adopting the term entropy in information theory. There are relationships between thermodynamic and informational entropy. For example, Maxwell's demon reverses thermodynamic entropy with information but getting that information exactly balances out the thermodynamic gain the demon would otherwise achieve.

It is important to remember that entropy is a quantity defined in the context of a probabilistic model for a data source. Independent fair coin flips have an entropy of 1 bit per flip. A source that always generates a long string of A's has an entropy of 0, since the next character will always be an 'A'. Empirically, it seems that entropy of English text is about 1.5 bits per character (Try compressing it with the PPM compression algorithm!), though clearly that will vary from text source to text source. The entropy rate of a data source means the average number of bits per symbol needed to encode it.

1. Many data bits may not convey information. For example, data structures often store information redundantly, or have identical sections regardless of the information in the data structure.
2. The amount of entropy is not always an integer number of bits.

Entropy effectively bounds the performance of the strongest non-lossy (or nearly non-lossy) compression possible, which can be realized in theory by using the typical set or in practice using Huffman, Lempel-Ziv or Arithmetic coding.

A common way to define entropy for text is based on the Markov model of text. For an order-0 source (each character is selected independent of the last characters), the entropy is:

Where is the probability of . For a second-order Markov source (one in which probabilities are dependent on the preceding character), the entropy rate is:

Where is a state (certain preceding characters) and is the probability of given as the previous character (s).

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