# Information theory

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Information theory is the mathematical study of the quantification, storage, and communication of information. The field was established and put on a firm footing by Claude Shannon in the 1940s, [1] though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, neurobiology, physics, and electrical engineering. [2]

## Contents

A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a die (which has six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory and information-theoretic security.

Applications of fundamental topics of information theory include source coding/data compression (e.g. for ZIP files), and channel coding/error detection and correction (e.g. for DSL). Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones and the development of the Internet. [2] The theory has found applications in other areas, including statistical inference, [3] cryptography, neurobiology, [4] perception, [5] linguistics, the evolution [6] and function [7] of molecular codes (bioinformatics), thermal physics, [8] molecular dynamics, [9] quantum computing, black holes, information retrieval, intelligence gathering, plagiarism detection, [10] pattern recognition, anomaly detection, [11] imaging system design, [12] epistemology, [13] and even art creation.

## Overview

Information theory studies the transmission, processing, extraction, and utilization of information. Abstractly, information can be thought of as the resolution of uncertainty. In the case of communication of information over a noisy channel, this abstract concept was formalized in 1948 by Claude Shannon in a paper entitled A Mathematical Theory of Communication , in which information is thought of as a set of possible messages, and the goal is to send these messages over a noisy channel, and to have the receiver reconstruct the message with low probability of error, in spite of the channel noise. Shannon's main result, the noisy-channel coding theorem, showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent. [4]

Coding theory is concerned with finding explicit methods, called codes, for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity. 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.[ citation needed ]

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, such as the unit ban.[ citation needed ]

## Historical background

The landmark event establishing the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948. He came to be known as the "father of information theory". [14] [15] [16] Shannon outlined some of his initial ideas of information theory as early as 1939 in a letter to Vannevar Bush. [16]

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 (recalling the Boltzmann constant), 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 one sequence of symbols from any other, thus quantifying information as H = log Sn = n log S, where S was the number of possible symbols, and n the number of symbols in a transmission. The unit of information was therefore the decimal digit, which since has sometimes been called the hartley in his honor 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.[ citation needed ]

Much of the mathematics behind information theory with events of different probabilities were 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 .[ citation needed ]

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:

"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; as well as
• the bit—a new way of seeing the most fundamental unit of information.[ citation needed ]

## Quantities of information

Information theory is based on probability theory and statistics, where quantified information is usually described in terms of bits. Information theory often concerns itself with measures of information of the distributions associated with random variables. One of the most important measures is called entropy, which forms the building block of many other measures. Entropy allows quantification of measure of information in a single random variable. [17] Another useful concept is mutual information defined on two random variables, which describes the measure of information in common between those variables, which can be used to describe their correlation. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution.

The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A 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 decimal digit, 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 = 0. This is justified because ${\displaystyle \lim _{p\rightarrow 0+}p\log p=0}$ for any logarithmic base.

### Entropy of an information source

Based on the probability mass function of each source symbol to be communicated, the Shannon entropy H, in units of bits (per symbol), is given by

${\displaystyle H=-\sum _{i}p_{i}\log _{2}(p_{i})}$

where pi is the probability of occurrence of the i-th possible value of the source symbol. This equation gives the entropy in the units of "bits" (per symbol) because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called the shannon in his honor. Entropy is also commonly computed using the natural logarithm (base , where e is Euler's number), which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas. Other bases are also possible, but less commonly used. For example, a logarithm of base 28 = 256 will produce a measurement in bytes per symbol, and a logarithm of base 10 will produce a measurement in decimal digits (or hartleys) per symbol.

Intuitively, the entropy HX of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known.

The entropy of a source that emits a sequence of N symbols that are independent and identically distributed (iid) is NH bits (per message of N symbols). If the source data symbols are identically distributed but not independent, the entropy of a message of length N will be less than NH.

If one transmits 1000 bits (0s and 1s), and the value of each of these bits is known to the receiver (has a specific value with certainty) ahead of transmission, it is clear that no information is transmitted. If, however, each bit is independently equally likely to be 0 or 1, 1000 shannons of information (more often called bits) have been transmitted. Between these two extremes, information can be quantified as follows. If ${\displaystyle \mathbb {X} }$ is the set of all messages {x1, ..., xn} that X could be, and p(x) is the probability of some ${\displaystyle x\in \mathbb {X} }$, then the entropy, H, of X is defined: [18]

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

(Here, I(x) is the self-information, which is the entropy contribution of an individual message, and ${\displaystyle \mathbb {E} _{X}}$ is the expected value.) A property of entropy is that it is maximized when all the messages in the message space are equiprobable p(x) = 1/n; i.e., most unpredictable, in which case H(X) = log n.

The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon (Sh) as unit:

${\displaystyle H_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p).}$

### 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—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.

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

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: [19]

${\displaystyle H(X|Y)=\mathbb {E} _{Y}[H(X|y)]=-\sum _{y\in Y}p(y)\sum _{x\in X}p(x|y)\log p(x|y)=-\sum _{x,y}p(x,y)\log p(x|y).}$

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:

${\displaystyle H(X|Y)=H(X,Y)-H(Y).\,}$

### 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:

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

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

A basic property of the mutual information is that

${\displaystyle I(X;Y)=H(X)-H(X|Y).\,}$

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:

${\displaystyle I(X;Y)=I(Y;X)=H(X)+H(Y)-H(X,Y).\,}$

Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) between the posterior probability distribution of X given the value of Y and the prior distribution on X:

${\displaystyle I(X;Y)=\mathbb {E} _{p(y)}[D_{\mathrm {KL} }(p(X|Y=y)\|p(X))].}$

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:

${\displaystyle I(X;Y)=D_{\mathrm {KL} }(p(X,Y)\|p(X)p(Y)).}$

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 ${\displaystyle p(X)}$, and an arbitrary probability distribution ${\displaystyle q(X)}$. If we compress data in a manner that assumes ${\displaystyle q(X)}$ is the distribution underlying some data, when, in reality, ${\displaystyle p(X)}$ is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus defined

${\displaystyle D_{\mathrm {KL} }(p(X)\|q(X))=\sum _{x\in X}-p(x)\log {q(x)}\,-\,\sum _{x\in X}-p(x)\log {p(x)}=\sum _{x\in X}p(x)\log {\frac {p(x)}{q(x)}}.}$

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

Another interpretation of the KL divergence is the "unnecessary surprise" introduced by a prior from the truth: suppose a number X is about to be drawn randomly from a discrete set with probability distribution ${\displaystyle p(x)}$. If Alice knows the true distribution ${\displaystyle p(x)}$, while Bob believes (has a prior) that the distribution is ${\displaystyle q(x)}$, then Bob will be more surprised than Alice, on average, upon seeing the value of X. The KL divergence is the (objective) expected value of Bob's (subjective) surprisal minus Alice's surprisal, measured in bits if the log is in base 2. In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him.

### Directed Information

Directed information , ${\displaystyle I(X^{n}\to Y^{n})}$, is an information theory measure that quantifies the information flow from the random process ${\displaystyle X^{n}=\{X_{1},X_{2},\dots ,X_{n}\}}$ to the random process ${\displaystyle Y^{n}=\{Y_{1},Y_{2},\dots ,Y_{n}\}}$. The term directed information was coined by James Massey and is defined as

${\displaystyle I(X^{n}\to Y^{n})\triangleq \sum _{i=1}^{n}I(X^{i};Y_{i}|Y^{i-1})}$,

where ${\displaystyle I(X^{i};Y_{i}|Y^{i-1})}$ is the conditional mutual information ${\displaystyle I(X_{1},X_{2},...,X_{i};Y_{i}|Y_{1},Y_{2},...,Y_{i-1})}$.

In contrast to mutual information, directed information is not symmetric. The ${\displaystyle I(X^{n}\to Y^{n})}$ measures the information bits that are transmitted causally[definition of causal transmission?] from ${\displaystyle X^{n}}$ to ${\displaystyle Y^{n}}$. The Directed information has many applications in problems where causality plays an important role such as capacity of channel with feedback, [20] [21] capacity of discrete memoryless networks with feedback, [22] gambling with causal side information, [23] compression with causal side information, [24] and in real-time control communication settings, [25] [26] statistical physics. [27]

### Other quantities

Other important information theoretic quantities include the Rényi entropy and the Tsallis entropy (generalizations of the concept of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information. Also, pragmatic information has been proposed as a measure of how much information has been used in making a decision.

## 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:
• 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.

### 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 less restrictive 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

${\displaystyle r=\lim _{n\to \infty }H(X_{n}|X_{n-1},X_{n-2},X_{n-3},\ldots );}$

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

${\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}H(X_{1},X_{2},\dots X_{n});}$

that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result. [28]

Information rate is defined as

${\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}I(X_{1},X_{2},\dots X_{n};Y_{1},Y_{2},\dots Y_{n});}$

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 is the primary motivation of information theory. However, channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality.

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

${\displaystyle {\xrightarrow[{\text{Message}}]{W}}{\begin{array}{|c| }\hline {\text{Encoder}}\\f_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Encoded \atop sequence} }]{X^{n}}}{\begin{array}{|c| }\hline {\text{Channel}}\\p(y|x)\\\hline \end{array}}{\xrightarrow[{\mathrm {Received \atop sequence} }]{Y^{n}}}{\begin{array}{|c| }\hline {\text{Decoder}}\\g_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Estimated \atop message} }]{\hat {W}}}}$

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:

${\displaystyle C=\max _{f}I(X;Y).\!}$

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.

#### Capacity of particular channel models

• 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 Hb(p) bits per channel use, where Hb is the binary entropy function to the base-2 logarithm:
• 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.

#### Channels with memory and directed information

In practice many channels have memory. Namely, at time ${\displaystyle i}$ the channel is given by the conditional probability${\displaystyle P(y_{i}|x_{i},x_{i-1},x_{i-2},...,x_{1},y_{i-1},y_{i-2},...,y_{1}).}$. It is often more comfortable to use the notation ${\displaystyle x^{i}=(x_{i},x_{i-1},x_{i-2},...,x_{1})}$ and the channel become ${\displaystyle P(y_{i}|x^{i},y^{i-1}).}$. In such a case the capacity is given by the mutual information rate when there is no feedback available and the Directed information rate in the case that either there is feedback or not [20] [29] (if there is no feedback the directed information equals the mutual information).

### Fungible information

Fungible information is the information for which the means of encoding is not important. [30] Classical information theorists and computer scientists are mainly concerned with information of this sort. It is sometimes referred as speakable information. [31]

## Applications to other fields

### 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 World War II 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 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 random seeds external to the software 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.

### Seismic exploration

One early commercial application of information theory was in the field of seismic oil exploration. Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods. [32]

### Semiotics

Semioticians Doede Nauta and Winfried Nöth both considered Charles Sanders Peirce as having created a theory of information in his works on semiotics. [33] :171 [34] :137 Nauta defined semiotic information theory as the study of "the internal processes of coding, filtering, and information processing." [33] :91

Concepts from information theory such as redundancy and code control have been used by semioticians such as Umberto Eco and Ferruccio Rossi-Landi to explain ideology as a form of message transmission whereby a dominant social class emits its message by using signs that exhibit a high degree of redundancy such that only one message is decoded among a selection of competing ones. [35]

### Integrated process organization of neural information

Quantitative information theoretic methods have been applied in cognitive science to analyze the integrated process organization of neural information in the context of the binding problem in cognitive neuroscience. [36] In this context, either an information-theoretical measure, such as functional clusters (Gerald Edelman and Giulio Tononi's functional clustering model and dynamic core hypothesis (DCH) [37] ) or effective information (Tononi's integrated information theory (IIT) of consciousness [38] [39] [40] ), is defined (on the basis of a reentrant process organization, i.e. the synchronization of neurophysiological activity between groups of neuronal populations), or the measure of the minimization of free energy on the basis of statistical methods (Karl J. Friston's free energy principle (FEP), an information-theoretical measure which states that every adaptive change in a self-organized system leads to a minimization of free energy, and the Bayesian brain hypothesis [41] [42] [43] [44] [45] ).

### Miscellaneous applications

Information theory also has applications in the search for extraterrestrial intelligence, [46] black holes, bioinformatics [ citation needed ], and gambling.

## Related Research Articles

In computer science and information theory, a Huffman code is a particular type of optimal prefix code that is commonly used for lossless data compression. The process of finding or using such a code is Huffman coding, an algorithm developed by David A. Huffman while he was a Sc.D. student at MIT, and published in the 1952 paper "A Method for the Construction of Minimum-Redundancy Codes".

In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable , which takes values in the set and is distributed according to , the entropy is where denotes the sum over the variable's possible values. The choice of base for , the logarithm, varies for different applications. Base 2 gives the unit of bits, while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable.

Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.

In the field of data compression, Shannon–Fano coding, named after Claude Shannon and Robert Fano, is one of two related techniques for constructing a prefix code based on a set of symbols and their probabilities.

A binary symmetric channel is a common communications channel model used in coding theory and information theory. In this model, a transmitter wishes to send a bit, and the receiver will receive a bit. The bit will be "flipped" with a "crossover probability" of p, and otherwise is received correctly. This model can be applied to varied communication channels such as telephone lines or disk drive storage.

Rate–distortion theory is a major branch of information theory which provides the theoretical foundations for lossy data compression; it addresses the problem of determining the minimal number of bits per symbol, as measured by the rate R, that should be communicated over a channel, so that the source can be approximately reconstructed at the receiver without exceeding an expected distortion D.

Channel capacity, in electrical engineering, computer science, and information theory, is the theoretical maximum rate at which information can be reliably transmitted over a communication channel.

In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. That this set has total probability close to one is a consequence of the asymptotic equipartition property (AEP) which is a kind of law of large numbers. The notion of typicality is only concerned with the probability of a sequence and not the actual sequence itself.

In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.

In information theory, the information content, self-information, surprisal, or Shannon information is a basic quantity derived from the probability of a particular event occurring from a random variable. It can be thought of as an alternative way of expressing probability, much like odds or log-odds, but which has particular mathematical advantages in the setting of information theory.

In mathematical statistics, the Kullback–Leibler (KL) divergence, denoted , is a type of statistical distance: a measure of how one probability distribution P is different from a second, reference probability distribution Q. Mathematically, it is defined as

In information theory, Shannon's source coding theorem establishes the statistical limits to possible data compression for data whose source is an independent identically-distributed random variable, and the operational meaning of the Shannon entropy.

In information theory, redundancy measures the fractional difference between the entropy H(X) of an ensemble X, and its maximum possible value . Informally, it is the amount of wasted "space" used to transmit certain data. Data compression is a way to reduce or eliminate unwanted redundancy, while forward error correction is a way of adding desired redundancy for purposes of error detection and correction when communicating over a noisy channel of limited capacity.

In information theory, the noisy-channel coding theorem, establishes that for any given degree of noise contamination of a communication channel, it is possible to communicate discrete data nearly error-free up to a computable maximum rate through the channel. This result was presented by Claude Shannon in 1948 and was based in part on earlier work and ideas of Harry Nyquist and Ralph Hartley.

In information theory, information dimension is an information measure for random vectors in Euclidean space, based on the normalized entropy of finely quantized versions of the random vectors. This concept was first introduced by Alfréd Rényi in 1959.

The decisive event which 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 1948.

The mathematical theory of information is based on probability theory and statistics, and measures information with several quantities of information. 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, or more correctly the shannon, based on the binary logarithm. Although "bit" is more frequently used in place of "shannon", its name is not distinguished from the bit as used in data-processing to refer to a binary value or stream regardless of its entropy Other units include the nat, based on the natural logarithm, and the hartley, based on the base 10 or common logarithm.

Inequalities are very important in the study of information theory. There are a number of different contexts in which these inequalities appear.

Asymmetric numeral systems (ANS) is a family of entropy encoding methods introduced by Jarosław (Jarek) Duda from Jagiellonian University, used in data compression since 2014 due to improved performance compared to previous methods. ANS combines the compression ratio of arithmetic coding, with a processing cost similar to that of Huffman coding. In the tabled ANS (tANS) variant, this is achieved by constructing a finite-state machine to operate on a large alphabet without using multiplication.

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### Other journal articles

• J. L. Kelly Jr., Princeton, "A New Interpretation of Information Rate" Bell System Technical Journal, Vol. 35, July 1956, pp. 917–26.
• R. Landauer, IEEE.org, "Information is Physical" Proc. Workshop on Physics and Computation PhysComp'92 (IEEE Comp. Sci.Press, Los Alamitos, 1993) pp. 1–4.
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