Information theory

Not to be confused with information technology, information science, or informatics. Information technology ( IT) as defined by the Information Technology Association of America (ITAA is "the study design development implementation support Information science is an interdisciplinary science primarily concerned with the collection classification, manipulation storage retrieval and dissemination Informatics is the science of Information, the practice of Information processing, and the engineering of Information systems.

Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of Information as a concept has a diversity of meanings from everyday usage to technical settings Historically, information theory was developed to find fundamental limits on compressing and reliably communicating data. Communication is the process of conveying information from a sender to a receiver with the use of a medium in which the communicated information is understood the same way 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,^[1] the evolution^[2] and function^[3] of molecular codes, model selection^[4] in ecology, thermal physics,^[5] quantum computing, plagiarism detection^[6] and other forms of data analysis. Inferential statistics or statistical induction comprises the use of Statistics to make Inferences concerning some unknown aspect of a Population Natural language processing ( NLP) is a subfield of Artificial intelligence and Computational linguistics. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Neurobiology is the study of cells of the Nervous system and the organization of these cells into functional circuits that process information and mediate behavior A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition Data analysis is the process of looking at and summarizing Data with the intent to extract useful Information and develop conclusions ^[7]

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. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way 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. Lossless data compression is a class of Data compression Algorithms that allows the exact original data to be reconstructed from the compressed data g. ZIP files), lossy data compression (e. The ZIP File format is a Data compression and archival format. A lossy compression method is one where compressing data and then decompressing it retrieves data that may well be different from the original but is close enough to be useful g. MP3s), and channel coding (e. MPEG-1 Audio Layer 3, more commonly referred to as MP3, is a Digital audio encoding format using a form of Lossy data compression In Electrical engineering, Computer science and Information theory, channel capacity is the tightest upper bound on the amount of Information g. for DSL lines). The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Neurobiology is the study of cells of the Nervous system and the organization of these cells into functional circuits that process information and mediate behavior Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of 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. See also Voyager 1 and Voyager 2. The Voyager program consists of a pair of unmanned scientific probes Voyager 1 and The Internet is a global system of interconnected Computer networks Linguistics is the scientific study of Language, encompassing a number of sub-fields A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e 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. In Electrical engineering, Computer science and Information theory, channel capacity is the tightest upper bound on the amount of Information 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. Claude Elwood Shannon (April 30 1916 – February 24 2001 an American Electronic engineer and Mathematician, is "the father of Information "A Mathematical Theory of Communication" is an influential 1948 article by Mathematician Claude E " 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. In Information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the limits to possible Data compression, and the operational In Information theory, the noisy-channel coding theorem establishes that however contaminated with noise interference a communication channel may be it is possible to communicate 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. An adaptive system is a System that is able to adapt its behavior according to changes in its environment or in parts of the system itself In Artificial intelligence, anticipation is the concept of an agent making decisions based on predictions expectations or beliefs about the future This article describes complex system as a type of system For other meanings see Complex systems. This article describes complex systems as field of Science. For other meanings see Complex system. Cybernetics is the interdisciplinary study of the Structure of Complex systems especially Communication processes control mechanisms and Feedback Informatics is the science of Information, the practice of Information processing, and the engineering of Information systems. Machine learning is a subfield of Artificial intelligence that is concerned with the design and development of Algorithms and techniques that allow computers to "learn" Systems science is the Interdisciplinary field of science which studies the nature of Complex systems in Nature, Society, and Science 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 one of the most important and direct applications of Information 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 Mathematics, Computer science, Telecommunication, and Information theory, error detection and correction has great practical importance in 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). In Cryptography, a code is a method used to transform a Message into an obscured form preventing those who do not possess special information or key In Cryptography, a cipher (or cypher) is an Algorithm for performing Encryption and Decryption &mdash a series of well-defined steps Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Cryptanalysis (from the Greek kryptós, "hidden" and analýein, "to loosen" or "to untie" is the study of methods for See the article ban (information) for a historical application. A ban, sometimes called a hartley (symbol Hart) or a dit (abbreviation of d ecimal dig' it') is a Logarithmic unit which

Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition. Information retrieval ( IR) is the science of searching for documents for Information within documents and for metadata about documents as well as that Intelligence (abbreviated int or intel) is not Information, but the product of evaluated information valued for its currency and relevance rather than Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Musical composition is an original piece of Music the structure of a musical piece the process of creating a new

Historical background

Main article: History of information theory

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. The decisive event which established the discipline of Information theory, and brought it to immediate worldwide attention was the publication of Claude E Claude Elwood Shannon (April 30 1916 – February 24 2001 an American Electronic engineer and Mathematician, is "the father of Information "A Mathematical Theory of Communication" is an influential 1948 article by Mathematician Claude E Bell Labs Technical Journal is the in-house journal for scientists of Bell Labs / Alcatel-Lucent.

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. Harry Nyquist ( né Harry Theodor Nyqvist pron, not as often pronounced ( February 7, 1889 – April 4, 1976) was an important 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. Ralph Vinton Lyon Hartley ( November 30, 1888 – May 1, 1970) was an Electronics researcher 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. A ban, sometimes called a hartley (symbol Hart) or a dit (abbreviation of d ecimal dig' it') is a Logarithmic unit which Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers. Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician The Enigma machines were a family of portable Cipher machines

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. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " Ludwig Eduard Boltzmann ( February 20, 1844 &ndash September 5, 1906) was an Austrian Physicist famous for his founding Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist 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. Rolf Landauer (1927–1999 was an IBM Physicist who in 1961 demonstrated that when Information is lost in an Irreversible circuit, the There are close parallels between the mathematical expressions for the thermodynamic Entropy, usually denoted by S, of a physical system in the Statistical thermodynamics

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

Main article: Quantities of information

Information theory is based on probability theory and statistics. Redundancy in Information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message In Information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the limits to possible Data compression, and the operational In Probability theory and Information theory, the mutual information (sometimes known by the archaic term transinformation) of two Random In Electrical engineering, Computer science and Information theory, channel capacity is the tightest upper bound on the amount of Information In Information theory, the noisy-channel coding theorem establishes that however contaminated with noise interference a communication channel may be it is possible to communicate In Information theory, the Shannon–Hartley theorem is an application of the Noisy channel coding theorem to the archetypal case of a continuous-time analog communications A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication The mathematical theory of information is based on Probability theory and Statistics, and measures information with several quantities of information. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. 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. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Probability theory and Information theory, the mutual information (sometimes known by the archaic term transinformation) of two Random 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. Channel, in communications (sometimes called communications channel) refers to the medium used to convey Information from a

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. A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication In Mathematics, the binary logarithm (log2 n) is the Logarithm for Base 2 Other units include the nat, which is based on the natural logarithm, and the hartley, which is based on the common logarithm. A nat (sometimes also nit or even nepit) is a Logarithmic unit of Information or entropy, based on Natural logarithms and The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational A ban, sometimes called a hartley (symbol Hart) or a dit (abbreviation of d ecimal dig' it') is a Logarithmic unit which The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base

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

Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function,

H b (p)

. In the theory of Probability and Statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes "success" In Information theory, the binary entropy function, denoted H(p \ or H_{\mathrm b}(p \ is defined as the entropy of a Bernoulli The entropy is maximized at 1 bit per trial when the two possible outcomes are equally probable, as in an unbiased coin toss.

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:^[8]

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

(Here, $I (x)$ is the self-information, which is the entropy contribution of an individual message. In Information theory (elaborated by Claude E Shannon, 1948) self-information is a measure of the information content associated with the outcome ) 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: $(X, Y)$ . In Information theory, the binary entropy function, denoted H(p \ or H_{\mathrm b}(p \ is defined as the entropy of a Bernoulli The joint entropy is an entropy measure used in Information theory. This implies that if $X$ and $Y$ are independent, then their joint entropy is the sum of their individual entropies. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other

For example, if $(X, Y)$ represents the position of a chess piece — $X$ 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. Chess is a recreational and competitive Game played between two players.

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

Despite similar notation, joint entropy should not be confused with cross entropy. In Information theory, the cross entropy between two Probability distributions measures the average number of Bits needed to identify an event from a set

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 $Y$ :^[9]

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

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. In Information theory, the conditional entropy (or equivocation) quantifies the remaining entropy (i 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. In Probability theory and Information theory, the mutual information (sometimes known by the archaic term transinformation) of two Random 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 $S I$ (Specific mutual Information) is the pointwise mutual information. Pointwise mutual information (PMI (or specific mutual information) is a measure of association used in Information theory and Statistics.

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. In Mathematics, the term "symmetric function" can mean two different things In Probability theory and Information theory, the Kullback–Leibler divergence (also information divergence, information gain, or relative The posterior probability of a Random event or an uncertain proposition is the Conditional probability that is assigned after the relevant evidence is taken A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. 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. The likelihood ratio often denoted by \Lambda (the capital Greek letter Lambda) is the ratio of the maximum Probability of a Result In Probability theory, the multinomial distribution is a generalization of the Binomial distribution. Pearson's chi-square (&chi2 test is the best-known of several Chi-square tests – statistical procedures whose results are evaluated by reference

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). In Probability theory and Information theory, the Kullback–Leibler divergence (also information divergence, information gain, or relative In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable 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). In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater

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. In Information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity uncertainty or randomness Differential entropy (also referred to as continuous entropy) is a concept in Information theory which tries to extend the idea of (Shannon entropy )

Coding theory

Main article: Coding theory

A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction.

A picture showing scratches on the readable surface of a CD-R. Coding theory is one of the most important and direct applications of Information theory. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction. In Mathematics, Computer science, Telecommunication, and Information theory, error detection and correction has great practical importance in

Coding theory is one of the most important and direct applications of information 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. In Mathematics, Computer science, Telecommunication, and Information theory, error detection and correction has great practical importance in 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:

lossless data compression: the data must be reconstructed exactly;
lossy data compression: allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. Lossless data compression is a class of Data compression Algorithms that allows the exact original data to be reconstructed from the compressed data A lossy compression method is one where compressing data and then decompressing it retrieves data that may well be different from the original but is close enough to be useful This subset of Information theory is called rate–distortion theory. Rate–distortion theory is a major branch of Information theory which provides the theoretical foundations for Lossy data compression; it addresses the problem of

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. Redundancy in Information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message e. error correction) needed to transmit the data efficiently and faithfully across a noisy channel. In Mathematics, Computer science, Telecommunication, and Information theory, error detection and correction has great practical importance in

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. In Information theory, a relay channel is a Probability model on the Communication between a Sender and a receiver aided by one or more intermediate A computer network is a group of interconnected Computers. Networks may be classified according to a wide variety of characteristics 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 source or sender is one of the basic concepts of Communication and Information processing. 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. "IID" or "iid" redirects here For other uses see IID (disambiguation. Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems In the mathematical sciences, a stationary process (or strict(ly stationary process or strong(ly stationary process) is a Stochastic process All such sources are stochastic. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. These terms are well studied in their own right outside information theory.

Rate

Information rate is the average entropy per symbol. The entropy rate of a Stochastic process is informally the time density of the average information in a stochastic process 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. ^[10]

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. Redundancy in Information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message

Channel capacity

Main article: Noisy channel coding theorem

Communications over a channel—such as an ethernet wire—is the primary motivation of information theory. In Information theory, the noisy-channel coding theorem establishes that however contaminated with noise interference a communication channel may be it is possible to communicate Ethernet is a family of frame -based Computer networking technologies for Local area networks (LANs 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. Conditional probability is the Probability of some event A, given the occurrence of some other event B. 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). In Science, and especially in Physics and Telecommunication, noise is fluctuations in and the addition of external factors to the stream of target 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. In the fields of communications, Signal processing, and in Electrical engineering more generally a signal is any time-varying or spatial-varying quantity 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). In Probability theory and Information theory, the mutual information (sometimes known by the archaic term transinformation) of two Random In Electrical engineering, Computer science and Information theory, channel capacity is the tightest upper bound on the amount of Information 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. In Computer science, a channel code is a broadly used term mostly referring to the Forward error correction code and Bit interleaving in communication and In Mathematics, Computer science, Telecommunication, and Information theory, error detection and correction has great practical importance in

Channel capacity of particular model channels

A continuous-time analog communications channel subject to Gaussian noise — see Shannon–Hartley theorem. In Information theory, the Shannon–Hartley theorem is an application of the Noisy channel coding theorem to the archetypal case of a continuous-time analog communications

A binary symmetric channel (BSC) with crossover probability p is a binary input, binary output channel that flips the input bit with probability p. A binary symmetric channel (or BSC is a common Communications channel model used in Coding theory and Information theory. The BSC has a capacity of $1 - H b (p)$ bits per channel use, where $H b$ is the binary entropy function:

A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. In Information theory, the binary entropy function, denoted H(p \ or H_{\mathrm b}(p \ is defined as the entropy of a Bernoulli 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.

Applications to other fields

Intelligence uses and secrecy applications

Information theoretic concepts apply to cryptography and cryptanalysis. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Cryptanalysis (from the Greek kryptós, "hidden" and analýein, "to loosen" or "to untie" is the study of methods for 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. A ban, sometimes called a hartley (symbol Hart) or a dit (abbreviation of d ecimal dig' it') is a Logarithmic unit which ULTra ("Urban Light Transport" is a Personal rapid transit system from Advanced Transport Systems Ltd a company based in Cardiff, Wales. The Enigma machine is any one of a family of related electro-mechanical Rotor machines used to generate Ciphers for the Encryption and decryption of Victory in Europe Day ( V-E Day or VE Day) was May 7 and May 8, 1945, the dates when the World War II Allies Shannon himself defined an important concept now called the unicity distance. Unicity distance is a term used in Cryptography referring to the length of an original Ciphertext needed to break the cipher by reducing the number of possible Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability. Redundancy in Information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message In Cryptography, plaintext is the information which the sender wishes to transmit to the receiver(s

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. In Cryptanalysis, a brute force attack is a method of defeating a Cryptographic scheme by trying a large number of possibilities for example possible keys Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key Symmetric-key algorithms are a class of Algorithms for Cryptography that use trivially related often identical Cryptographic keys for both decryption In Cryptography, a block cipher is a symmetric key Cipher which operates on fixed-length groups of Bits termed blocks, with an 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. A Cryptosystem is information-theoretically secure if its security derives purely from Information theory. In Cryptography, the one-time pad (OTP is an Encryption Algorithm where the Plaintext is combined with a random key or "pad" 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 Probability theory and Information theory, the mutual information (sometimes known by the archaic term transinformation) of two Random In Cryptography, plaintext is the information which the sender wishes to transmit to the receiver(s In Cryptography, a key is a piece of information (a Parameter) that determines the functional output of a cryptographic algorithm 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. The Venona project was a long-running and highly secret collaboration between Intelligence agencies of the United States and United Kingdom that involved The Union of Soviet Socialist Republics (USSR was a constitutionally Socialist state that existed in Eurasia from 1922 to 1991

Pseudorandom number generation

Pseudorandom number generators are widely available in computer language libraries and application programs. A pseudorandom number generator ( PRNG) is an Algorithm for generating a sequence of numbers that approximates the properties of random numbers 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. A cryptographically secure pseudo-random number generator ( CSPRNG) is a Pseudo-random number generator (PRNG with properties that make it suitable for use in A random seed (or seed state, or just seed) is a Number (or vector) used to initialize a Pseudorandom number generator. These can be obtained via extractors, if done carefully. This article is about the extractor in mathematics for other usage of this word see Extractor (firearms. 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. In Probability theory or Information theory, the min-entropy of a discrete random event x with possible states (or outcomes 1 In Information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity uncertainty or randomness 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. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way

Miscellaneous applications

Information theory also has applications in gambling and investing, black holes, bioinformatics, and music. Statistical inference might be thought of as gambling theory applied to the world around The black hole information paradox results from the combination of Quantum mechanics and General relativity. Bioinformatics is the application of information technology to the field of molecular biology Music is an Art form in which the medium is Sound organized in Time.

References

Footnotes

^ F. Rieke, D. Warland, R Ruyter van Steveninck, W Bialek, Spikes: Exploring the Neural Code. The MIT press (1997).
^ cf. Huelsenbeck, J. P. , F. Ronquist, R. Nielsen and J. P. Bollback (2001) Bayesian inference of phylogeny and its impact on evolutionary biology, Science 294:2310-2314
^ Rando Allikmets, Wyeth W. Wasserman, Amy Hutchinson, Philip Smallwood, Jeremy Nathans, Peter K. Rogan, Thomas D. Schneider, Michael Dean (1998) Organization of the ABCR gene: analysis of promoter and splice junction sequences, Gene 215:1, 111-122
^ Burnham, K. P. and Anderson D. R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Second Edition (Springer Science, New York) ISBN 978-0-387-95364-9.
^ Jaynes, E. T. (1957) Information Theory and Statistical Mechanics, Phys. Rev. 106:620
^ Charles H. Bennett, Ming Li, and Bin Ma (2003) Chain Letters and Evolutionary Histories, Scientific American 288:6, 76-81
^ David R. Anderson (November 1, 2003). Some background on why people in the empirical sciences may want to better understand the information-theoretic methods (pdf). Retrieved on 2007-12-30. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1460 - Wars of the Roses: Battle of Wakefield. 1816 - The Treaty of St
^ Fazlollah M. Reza (1961, 1994). An Introduction to Information Theory. Dover Publications, Inc. , New York. ISBN 0-486-68210-2.
^ Robert B. Ash (1965, 1990). Information Theory. Dover Publications, Inc. . ISBN 0-486-66521-6.
^ Jerry D. Gibson (1998). Digital Compression for Multimedia: Principles and Standards. Morgan Kaufmann. ISBN 1558603697.

The classic work

Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27, pp. Claude Elwood Shannon (April 30 1916 – February 24 2001 an American Electronic engineer and Mathematician, is "the father of Information "A Mathematical Theory of Communication" is an influential 1948 article by Mathematician Claude E 379–423 & 623–656, July & October, 1948. PDF.
Notes and other formats.

Ludwig Boltzmann formally defined entropy in 1870. Ludwig Eduard Boltzmann ( February 20, 1844 &ndash September 5, 1906) was an Austrian Physicist famous for his founding 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

Textbooks on information theory

Claude E. Shannon, Warren Weaver. Claude Elwood Shannon (April 30 1916 – February 24 2001 an American Electronic engineer and Mathematician, is "the father of Information The Mathematical Theory of Communication. Univ of Illinois Press, 1949. ISBN 0-252-72548-4
Robert Gallager. Robert G Gallager (born May 29, 1931 in Philadelphia PA) is an American electrical engineer known for his work on Information theory 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 M Cover (born August 7, 1938 in San Bernardino California) is Professor jointly in the Departments of Electrical Engineering and 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. Imre Csiszár is a Hungarian mathematician with contributions to Information theory and Probability theory. 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. Fazlollah M Reza (born on January 1, 1915 in Rasht, Iran) is an Iranian university professor 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, [1];

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

External links

Gibbs, M. There is much discussion in the academic world of Communication as to what actually constitutes communication Computability Computability An introduction The philosophy of information (PI is the area of research that studies conceptual issues arising at the intersection of Computer science, Information technology, Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Cryptanalysis (from the Greek kryptós, "hidden" and analýein, "to loosen" or "to untie" is the study of methods for There are close parallels between the mathematical expressions for the thermodynamic Entropy, usually denoted by S, of a physical system in the Statistical thermodynamics Intelligence (abbreviated int or intel) is not Information, but the product of evaluated information valued for its currency and relevance rather than Cybernetics is the interdisciplinary study of the Structure of Complex systems especially Communication processes control mechanisms and Feedback The decisive event which established the discipline of Information theory, and brought it to immediate worldwide attention was the publication of Claude E A Timeline of events related to theory]] compression]] correcting code]]s and related subjects Claude Elwood Shannon (April 30 1916 – February 24 2001 an American Electronic engineer and Mathematician, is "the father of Information Ralph Vinton Lyon Hartley ( November 30, 1888 – May 1, 1970) was an Electronics researcher Professor Hubert P Yockey (b April 15, 1916) PhD is a Physicist and information theorist. Coding theory is one of the most important and direct applications of Information theory. Detection theory, or signal detection theory, is a means to quantify the ability to discern between signal and noise. Estimation theory is a branch of Statistics and Signal processing that deals with estimating the values of parameters based on measured/empirical data In Statistics and Information theory, the Fisher information (denoted \mathcal{I}(\theta is the Variance of the score. In Algorithmic information theory (a subfield of Computer science) the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov-Chaitin Classical Information theory goes back to Claude Shannon. It is a theory of information transmission looking at communication and storage In Mathematics and especially in Statistical inference, information geometry is the study of Probability and information by way of Measures in information theory Many of the formulas in information theory have separate versions for continuous and discrete cases i The logic of information, or the logical theory of information, considers the information content of logical signs and expressions along the lines initially developed Network coding is a field of Information theory and Coding theory and is a method of attaining maximum information flow in a network. Quantum information science concerns information science that depends on quantum effects in physics Semiotic information theory considers the Information content of signs and expressions as it is conceived within the semiotic or sign-relational The philosophy of information (PI is the area of research that studies conceptual issues arising at the intersection of Computer science, Information technology, In Information theory (elaborated by Claude E Shannon, 1948) self-information is a measure of the information content associated with the outcome The joint entropy is an entropy measure used in Information theory. In Information theory, the conditional entropy (or equivocation) quantifies the remaining entropy (i Redundancy in Information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message Channel, in communications (sometimes called communications channel) refers to the medium used to convey Information from a A source or sender is one of the basic concepts of Communication and Information processing. The receiver in Information theory is the receiving end of a Communication channel. In Information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity uncertainty or randomness In Cybernetics the term variety denotes the total number of distinct states of a System. In Probability theory and Information theory, the mutual information (sometimes known by the archaic term transinformation) of two Random Pointwise mutual information (PMI (or specific mutual information) is a measure of association used in Information theory and Statistics. Differential entropy (also referred to as continuous entropy) is a concept in Information theory which tries to extend the idea of (Shannon entropy In Probability theory and Information theory, the Kullback–Leibler divergence (also information divergence, information gain, or relative In Electrical engineering, Computer science and Information theory, channel capacity is the tightest upper bound on the amount of Information Unicity distance is a term used in Cryptography referring to the length of an original Ciphertext needed to break the cipher by reducing the number of possible A ban, sometimes called a hartley (symbol Hart) or a dit (abbreviation of d ecimal dig' it') is a Logarithmic unit which In Information theory, a covert channel is a parasitic Communications channel that draws bandwidth from another channel in order to transmit information An encoder is a device used to change a signal (such as a Bitstream) or Data into a Code. For the drum and bass musician see Decoder (artist A decoder is a device which does the reverse of an Encoder, undoing the , "Quantum Information Theory", Eprint
Schneider, T. , "Information Theory Primer", Eprint
Srinivasa, S. "A Review on Multivariate Mutual Information" PDF.
Challis, J. Lateral Thinking in Information Retrieval
Journal of Chemical Education, Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? Nonsense!
IEEE Information Theory Society and the review articles.
On-line textbook: Information Theory, Inference, and Learning Algorithms, by David MacKay - gives an entertaining and thorough introduction to Shannon theory, including state-of-the-art methods from coding theory, such as arithmetic coding, low-density parity-check codes, and Turbo codes. Arithmetic coding is a method for Lossless data compression. Normally a string of characters such as the words "hello there" is represented using a fixed number of In Information theory, a low-density parity-check code (LDPC code is an Error correcting code, a method of transmitting a message over a noisy transmission In Electrical engineering and Digital communications, turbo codes (originally in French Turbocodes) are a class of high-performance error correction
A good tutorial of Information Theory.

Dictionary

-noun

(mathematics) A branch of applied mathematics and engineering involving the quantification of information sent over a communication channel, disregarding the meaning of the sent messages, exemplified by the noisy-channel coding theorem.

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