Information theory

Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed to find fundamental limits on compressing and reliably communicating data. Since its inception it has broadened to find applications in many other areas, including statistical inference, natural language processing, cryptography generally, networks other than communication networks -- as in neurobiology, the evolution and function of molecular codes, model selection in ecology, thermal physics, quantum computing, plagiarism detection and other forms of data analysis.

A key measure of information in the theory is known as information entropy, which is usually expressed by the average number of bits needed for storage or communication. Intuitively, entropy quantifies the uncertainty involved when encountering a random variable. For example, a fair coin flip will have less entropy than a roll of a die.

Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s), and channel coding (e.g. for DSL lines). The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the CD, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields. Important sub-fields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, and measures of information.

Overview
The main concepts of information theory can be grasped by considering the most widespread means of human communication: language. Two important aspects of a good language are as follows: First, the most common words (e.g., "a", "the", "I") should be shorter than less common words (e.g., "benefit", "generation", "mediocre"), so that sentences will not be too long. Such a tradeoff in word length is analogous to data compression and is the essential aspect of source coding. Second, if part of a sentence is unheard or misheard due to noise -— e.g., a passing car -— the listener should still be able to glean the meaning of the underlying message. Such robustness is as essential for an electronic communication system as it is for a language; properly building such robustness into communications is done by channel coding. Source coding and channel coding are the fundamental concerns of information theory.

Note that these concerns have nothing to do with the importance of messages. For example, a platitude such as "Thank you; come again" takes about as long to say or write as the urgent plea, "Call an ambulance!" while clearly the latter is more important and more meaningful. Information theory, however, does not consider message importance or meaning, as these are matters of the quality of data rather than the quantity and readability of data, the latter of which is determined solely by probabilities.

Information theory is generally considered to have been founded in 1948 by Claude Shannon in his seminal work, "A Mathematical Theory of Communication." The central paradigm of classical information theory is the engineering problem of the transmission of information over a noisy channel. The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy; and Shannon's noisy-channel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold called the channel capacity. The channel capacity can be approached in practice by using appropriate encoding and decoding systems.

Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half century or more: adaptive systems, anticipatory systems, artificial intelligence, complex systems, complexity science, cybernetics, informatics, machine learning, along with systems sciences of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.

Coding theory is concerned with finding explicit methods, called codes, of increasing the efficiency and reducing the net error rate of data communication over a noisy channel to near the limit that Shannon proved is the maximum possible for that channel. These codes can be roughly subdivided into data compression (source coding) and error-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban (information) for a historical application.

Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition.

Historical background
The landmark event that established the discipline of information theory, and brought 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 of 1948.

Prior to this paper, limited information theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation $$W = K \log m$$, where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish that one sequence of symbols from any other, thus quantifying information as $$H = \log S^n = n \log S$$, where S was the number of possible symbols, and n the number of symbols in a transmission. The natural unit of information was therefore the decimal digit, much later renamed the hartley in his honour as a unit or scale or measure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.

Much of the mathematics behind information theory with events of different probabilities was developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory.

In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that
 * "The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."

With it came the ideas of
 * the information entropy and redundancy of a source, and its relevance through the source coding theorem;
 * the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem;
 * the practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; and of course
 * the bit—a new way of seeing the most fundamental unit of information

Ways of measuring information
Information theory is based on probability theory and statistics. The most important quantities of information are entropy, the information in a random variable, and mutual information, the amount of information in common between two random variables. The former quantity indicates how easily message data can be compressed while the latter can be used to find the communication rate across a channel.

The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. The most common unit of information is the bit, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the hartley, which is based on the common logarithm.

In what follows, an expression of the form $$p \log p \,$$ is considered by convention to be equal to zero whenever p is. This is justified because $$\lim_{p \rightarrow 0+} p \log p = 0$$ for any logarithmic base.

Entropy
The entropy, $$H$$, of a discrete random variable $$X$$ is a measure of the amount of uncertainty associated with the value of $$X$$.

Suppose one transmits 1000 bits (0s and 1s). If these bits are known ahead of transmission (to be a certain value with absolute probability), logic dictates that no information has been transmitted. If, however, each is equally and independently likely to be 0 or 1, 1000 bits (in the information theoretic sense) have been transmitted. Between these two extremes, information can be quantified as follows. If $$\mathbb{X}\,$$ is the set of all messages $$x$$ that $$X$$ could be, and $$p(x)$$ is the probability of $$X$$ given $$x$$, then the entropy of $$X$$ is defined:


 * $$ H(X) = \mathbb{E}_{X} [I(x)] = -\sum_{x \in \mathbb{X}} p(x) \log p(x).$$

(Here, $$I(x)$$ is the self-information, which is the entropy contribution of an individual message.) An important property of entropy is that it is maximized when all the messages in the message space are equiprobable—i.e., most unpredictable—in which case $$H(X) = \log |\mathbb{X}|.$$

The special case of information entropy for a random variable with two outcomes is the binary entropy function:


 * $$H_\mbox{b}(p) = - p \log p - (1-p)\log (1-p).\,$$

Joint entropy
The joint entropy of two discrete random variables $$X$$ and $$Y$$ is merely the entropy of their pairing: $$(X, Y)$$. This implies that if $$X$$ and $$Y$$ are independent, then their joint entropy is the sum of their individual entropies.

For example, if $$(X,Y)$$ represents the position of a chess piece &mdash; $$X$$ the row and $$Y$$ the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece.


 * $$H(X, Y) = \mathbb{E}_{X,Y} [-\log p(x,y)] = - \sum_{x, y} p(x, y) \log p(x, y) \,$$

Despite similar notation, joint entropy should not be confused with cross entropy.

Conditional entropy (equivocation)
The conditional entropy or conditional uncertainty of $$X$$ given random variable $$Y$$ (also called the equivocation of $$X$$ about $$Y$$) is the average conditional entropy over $$Y$$:


 * $$ H(X|Y) = \mathbb E_Y [H(X|y)] = -\sum_{y \in Y} p(y) \sum_{x \in X} p(x|y) \log p(x|y) = -\sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(y)}.$$

Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use. A basic property of this form of conditional entropy is that:


 * $$ H(X|Y) = H(X,Y) - H(Y) .\,$$

Mutual information (transinformation)
Mutual information measures the amount of information that can be obtained about one random variable by observing another. It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of $$X$$ relative to $$Y$$ is given by:


 * $$I(X;Y) = \mathbb{E}_{X,Y} [SI(x,y)] = \sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)\, p(y)}$$

where $$SI$$ (Specific mutual Information) is the pointwise mutual information.

A basic property of the mutual information is that
 * $$I(X;Y) = H(X) - H(X|Y).\,$$

That is, knowing Y, we can save an average of $$I(X; Y)$$ bits in encoding X compared to not knowing Y.

Mutual information is symmetric:
 * $$I(X;Y) = I(Y;X) = H(X) + H(Y) - H(X,Y).\,$$

Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) of the posterior probability distribution of X given the value of Y to the prior distribution on X:
 * $$I(X;Y) = \mathbb E_{p(y)} [D_{\mathrm{KL}}( p(X|Y=y) \| p(X) )].$$

In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y. This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution:
 * $$I(X; Y) = D_{\mathrm{KL}}(p(X,Y) \| p(X)p(Y)).$$

Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ2 test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution.

Kullback–Leibler divergence (information gain)
The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions: a "true" probability distribution p(X), and an arbitrary probability distribution q(X). If we compress data in a manner that assumes q(X) is the distribution underlying some data, when, in reality, p(X) is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus defined


 * $$D_{\mathrm{KL}}(p(X) \| q(X)) = \sum_{x \in X} -p(x) \log {q(x)} \, - \, \left( -p(x) \log {p(x)}\right) = \sum_{x \in X} p(x) \log \frac{p(x)}{q(x)}.$$

Although it is sometimes used as a 'distance metric', it is not a true metric since it is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric).

Other quantities
Other important information theoretic quantities include Rényi entropy (a generalization of entropy) and differential entropy (a generalization of quantities of information to continuous distributions.)

Coding theory


Coding theory is one of the most important and direct applications of information theory. It can be subdivided into source coding theory and channel coding theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source.


 * Data compression (source coding): There are two formulations for the compression problem:
 * 1) lossless data compression: the data must be reconstructed exactly;
 * 2) lossy data compression: allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. This subset of Information theory is called rate–distortion theory.


 * Error-correcting codes (channel coding): While data compression removes as much redundancy as possible, an error correcting code adds just the right kind of redundancy (i.e. error correction) needed to transmit the data efficiently and faithfully across a noisy channel.

This division of coding theory into compression and transmission is justified by the information transmission theorems, or source–channel separation theorems that justify the use of bits as the universal currency for information in many contexts. However, these theorems only hold in the situation where one transmitting user wishes to communicate to one receiving user. In scenarios with more than one transmitter (the multiple-access channel), more than one receiver (the broadcast channel) or intermediary "helpers" (the relay channel), or more general networks, compression followed by transmission may no longer be optimal. Network information theory refers to these multi-agent communication models.

Source theory
Any process that generates successive messages can be considered a source of information. A memoryless source is one in which each message is an independent identically-distributed random variable, whereas the properties of ergodicity and stationarity impose more general constraints. All such sources are stochastic. These terms are well studied in their own right outside information theory.

Rate
Information rate is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is


 * $$r = \lim_{n \to \infty} H(X_n|X_{n-1},X_{n-2},X_{n-3}, \ldots);$$

that is, the conditional entropy of a symbol given all the previous symbols generated. For the more general case of a process that is not necessarily stationary, the average rate is


 * $$r = \lim_{n \to \infty} \frac{1}{n} H(X_1, X_2, \dots X_n);$$

that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result.

It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose. The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding.

Channel capacity
Communications over a channel—such as an ethernet wire—is the primary motivation of information theory. As anyone who's ever used a telephone (mobile or landline) knows, however, such channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality. How much information can one hope to communicate over a noisy (or otherwise imperfect) channel?

Consider the communications process over a discrete channel. A simple model of the process is shown below:



Here X represents the space of messages transmitted, and Y the space of messages received during a unit time over our channel. Let $$p(y|x)$$ be the conditional probability distribution function of Y given X. We will consider $$p(y|x)$$ to be an inherent fixed property of our communications channel (representing the nature of the noise of our channel). Then the joint distribution of X and Y is completely determined by our channel and by our choice of $$f(x)$$, the marginal distribution of messages we choose to send over the channel. Under these constraints, we would like to maximize the rate of information, or the signal, we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by:
 * $$ C = \max_{f} I(X;Y).\! $$

This capacity has the following property related to communicating at information rate R (where R is usually bits per symbol). For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε; that is, it is always possible to transmit with arbitrarily small block error. In addition, for any rate R > C, it is impossible to transmit with arbitrarily small block error.

Channel coding is concerned with finding such nearly optimal codes that can be used to transmit data over a noisy channel with a small coding error at a rate near the channel capacity.

Channel capacity of particular model channels

 * A continuous-time analog communications channel subject to Gaussian noise — see Shannon–Hartley theorem.


 * A binary symmetric channel (BSC) with crossover probability p is a binary input, binary output channel that flips the input bit with probability  p. The BSC has a capacity of $$1 - H_\mbox{b}(p)$$ bits per channel use, where $$H_\mbox{b}$$ is the binary entropy function:


 * [[Image:Binary symmetric channel.svg]]


 * A binary erasure channel (BEC) with erasure probability  p  is a binary input, ternary output channel. The possible channel outputs are 0, 1, and a third symbol 'e' called an erasure. The erasure represents complete loss of information about an input bit. The capacity of the BEC is 1 - p bits per channel use.


 * [[Image:Binary erasure channel.svg]]

Intelligence uses and secrecy applications
Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of WWII in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.

Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms (sometimes called secret key algorithms), such as block ciphers. The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time.

Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks. In such cases, the positive conditional mutual information between the plaintext and ciphertext (conditioned on the key) can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications. In other words, an eavesdropper would not be able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key. However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material.

Pseudorandom number generation
Pseudorandom number generators are widely available in computer language libraries and application programs. They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software. A class of improved random number generators is termed Cryptographically secure pseudorandom number generators, but even they require external to the software random seeds to work as intended. These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractors is min-entropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptography uses.

Miscellaneous applications
Information theory also has applications in gambling and investing, black holes, bioinformatics, and music.

The classic work

 * Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27, pp. 379–423 & 623–656, July & October, 1948. PDF. Notes and other formats.


 * Ludwig Boltzmann formally defined entropy in 1870. Compare: Boltzmann, Ludwig (1896, 1898). Vorlesungen über Gastheorie : 2 Volumes - Leipzig 1895/98 UB: O 5262-6. English version: Lectures on gas theory. Translated by Stephen G. Brush (1964) Berkeley: University of California Press; (1995) New York: Dover ISBN 0-486-68455-5

Other journal articles

 * R.V.L. Hartley, "Transmission of Information," Bell System Technical Journal, July 1928
 * J. L. Kelly, Jr., "A New Interpretation of Information Rate," Bell System Technical Journal, Vol. 35, July 1956, pp. 917-26
 * R. Landauer, Information is Physical Proc. Workshop on Physics and Computation PhysComp'92 (IEEE Comp. Sci.Press, Los Alamitos, 1993) pp. 1-4.
 * R. Landauer, "Irreversibility and Heat Generation in the Computing Process" IBM J. Res. Develop. Vol. 5, No. 3, 1961

Textbooks on information theory

 * Claude E. Shannon, Warren Weaver. The Mathematical Theory of Communication. Univ of Illinois Press, 1949. ISBN 0-252-72548-4
 * Robert Gallager. Information Theory and Reliable Communication. New York: John Wiley and Sons, 1968. ISBN 0-471-29048-3
 * Robert B. Ash. Information Theory. New York: Interscience, 1965. ISBN 0-470-03445-9. New York: Dover 1990. ISBN 0-486-66521-6
 * Thomas M. Cover, Joy A. Thomas. Elements of information theory, 1st Edition. New York: Wiley-Interscience, 1991. ISBN 0-471-06259-6.
 * 2nd Edition. New York: Wiley-Interscience, 2006. ISBN 0-471-24195-4.


 * Imre Csiszar, Janos Korner. Information Theory: Coding Theorems for Discrete Memoryless Systems Akademiai Kiado: 2nd edition, 1997. ISBN 9630574403
 * Raymond W. Yeung. A First Course in Information Theory Kluwer Academic/Plenum Publishers, 2002.  ISBN 0-306-46791-7
 * David J. C. MacKay. Information Theory, Inference, and Learning Algorithms Cambridge: Cambridge University Press, 2003. ISBN 0-521-64298-1
 * Stanford Goldman. Information Theory. New York: Prentice Hall, 1953. New York: Dover 1968 ISBN 0-486-62209-6, 2005 ISBN 0-486-44271-3
 * Fazlollah Reza. An Introduction to Information Theory. New York: McGraw-Hill 1961. New York: Dover 1994. ISBN 0-486-68210-2
 * Masud Mansuripur. Introduction to Information Theory. New York: Prentice Hall, 1987. ISBN 0-13-484668-0
 * Christoph Arndt: Information Measures, Information and its Description in Science and Engineering (Springer Series: Signals and Communication Technology), 2004, ISBN 978-3-540-40855-0, ;

Other books

 * Leon Brillouin, Science and Information Theory, Mineola, N.Y.: Dover, [1956, 1962] 2004. ISBN 0-486-43918-6
 * A. I. Khinchin, Mathematical Foundations of Information Theory, New York: Dover, 1957. ISBN 0-486-60434-9
 * H. S. Leff and A. F. Rex, Editors, Maxwell's Demon: Entropy, Information, Computing, Princeton University Press, Princeton, NJ (1990). ISBN 0-691-08727-X
 * Tom Siegfried, The Bit and the Pendulum, Wiley, 2000. ISBN 0-471-32174-5
 * Charles Seife, Decoding The Universe, Viking, 2006. ISBN 0-670-03441-X
 * Jeremy Campbell, Grammatical Man, Touchstone/Simon & Schuster, 1982, ISBN 0-671-44062-4

Applications

 * Cryptography
 * Cryptanalysis
 * Entropy in thermodynamics and information theory
 * Intelligence (information gathering)
 * Gambling
 * Cybernetics

History

 * History of information theory
 * Timeline of information theory
 * Shannon, C.E.
 * Hartley, R.V.L.
 * Yockey, H.P.

Theory

 * Coding theory
 * Source coding
 * Detection theory
 * Estimation theory
 * Fisher information
 * Kolmogorov complexity
 * Information Algebra
 * Information geometry
 * Information theory and measure theory
 * Logic of information
 * Network coding
 * Quantum information science
 * Semiotic information theory
 * Philosophy of Information

Concepts

 * Self-information
 * Information entropy
 * Joint entropy
 * Conditional entropy
 * Redundancy
 * Channel (communications)
 * Communication source
 * Receiver (information theory)
 * Rényi entropy
 * Variety
 * Mutual information
 * Pointwise Mutual Information (PMI)
 * Differential entropy
 * Kullback-Leibler divergence
 * Channel capacity
 * Unicity distance
 * ban (information)
 * Covert channel
 * Encoder
 * Decoder