Information theory

Information theory studies the, , and of. It was originally proposed by in 1948 to find fundamental limits on  and communication operations such as, in a landmark paper titled "". Its impact has been crucial to the success of the missions to deep space, the invention of the, the feasibility of s, the development of the , the study of  and of human perception, the understanding of s, and numerous other fields.

The field is at the intersection of, , , , , , and. The theory has also found applications in other areas, including, , , , human vision, the evolution and function of molecular codes , in statistics, , , , , , and. Important sub-fields of information theory include, , , , , and measures of information.

Applications of fundamental topics of information theory include (e.g. ),  (e.g. s and s), and  (e.g. for ). Information theory is used in, , , , and even in.

A key measure in information theory is "". Entropy quantifies the amount of uncertainty involved in the value of a or the outcome of a. For example, identifying the outcome of a fair (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a  (with six equally likely outcomes). Some other important measures in information theory are, , s, and.

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 made concrete in 1948 by in his paper "", in which "information" is thought of as a set of possible messages, where the goal is to send these messages over a noisy channel, and then to have the receiver reconstruct the message with low probability of error, in spite of the channel noise. Shannon's main result, the showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the, a quantity dependent merely on the statistics of the channel over which the messages are sent.

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 throughout the world over the past half century or more: s, s,, s, , , , , along with s of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of.

Coding theory 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. These codes can be roughly subdivided into (source coding) and  (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible.

A third class of information theory codes are cryptographic algorithms (both s and s). Concepts, methods and results from coding theory and information theory are widely used in and. See the article for a historical application.

Historical background
The landmark event that established the discipline of information theory and brought it to immediate worldwide attention was the publication of 's classic paper "" in the  in July and October 1948.

Prior to this paper, limited information-theoretic ideas had been developed at, all implicitly assuming events of equal probability. '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 ), 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. 's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one from any other, thus quantifying information as $H = log S^{n} = n log S$, where S was the number of possible symbols, and n the number of symbols in a transmission. The unit of information was therefore the, which has since sometimes been called the in his honor as a unit or scale or measure of information. in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war ciphers.

Much of the mathematics behind information theory with events of different probabilities were developed for the field of by  and. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by in the 1960s, are explored in .

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 and  of a source, and its relevance through the ;
 * the, and the of a noisy channel, including the promise of perfect loss-free communication given by the ;
 * the practical result of the for the channel capacity of a ; as well as
 * the —a new way of seeing the most fundamental unit of information.

Quantities of information
Information theory is based on and. Information theory often concerns itself with measures of information of the distributions associated with random variables. Important quantities of information are, a measure of information in a single , and , a measure of information in common between two random variables. 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. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy 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 of  that is used. A common unit of information is the, based on the. Other units include the, which is based on the , and the , which is based on the.

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 $$\lim_{p \rightarrow 0+} p \log p = 0$$ for any logarithmic base.

Entropy of an information source
Based on the of each source symbol to be communicated, the Shannon  $H$, in units of bits (per symbol), is given by
 * $$H = - \sum_{i} p_i \log_2 (p_i)$$

where $p_{i}$ 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 "" in his honor. Entropy is also commonly computed using the (base, where $e$ is ), which produces a measurement of entropy in "" 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 s per symbol, and a logarithm of base 10 will produce a measurement in decimal digits (or hartleys) per symbol.

Intuitively, the entropy $H_{X}$ 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 (iid) is $N ⋅ H$ 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 $N ⋅ H$.

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 𝕏 is the set of all messages $H_{b}(p)$ that $\{x_{1}, ..., x_{n}\}$ could be, and $X$ is the probability of some $$x \in \mathbb X$$, then the entropy, $p(x)$, of $H$ is defined:


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

(Here, $X$ is the, which is the entropy contribution of an individual message, and $I(x)$ is the .) A property of entropy is that it is maximized when all the messages in the message space are equiprobable $𝔼_{X}$; i.e., most unpredictable, in which case $p(x) = 1/n$.

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


 * $$H_{\mathrm{b}}(p) = - p \log_2 p - (1-p)\log_2 (1-p).$$

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

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


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

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

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


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

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:


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

Mutual information (transinformation)
 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 $Y$ relative to $X$ 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 $Y$ (Specific mutual Information) is the.

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 $SI$ bits in encoding X compared to not knowing Y.

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

Mutual information can be expressed as the average (information gain) between the  of X given the value of Y and the  on X:
 * $$I(X;Y) = \mathbb E_{p(y)} [D_{\mathrm{KL}}( p(X|Y=y) \| p(X) )].$$

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

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


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

Although it is sometimes used as a 'distance metric', KL divergence is not a true since it is not symmetric and does not satisfy the  (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 p(x). If Alice knows the true distribution p(x), while Bob believes (has a ) that the distribution is q(x), then Bob will be more than Alice, on average, upon seeing the value of X. The KL divergence is the (objective) expected value of Bob's (subjective) 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.

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

Coding theory
is one of the most important and direct applications of information theory. It can be subdivided into theory and  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:
 * the data must be reconstructed exactly;
 * allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. This subset of information theory is called .
 * Error-correcting codes (channel coding): While data compression removes as much as possible, an error correcting code adds just the right kind of redundancy (i.e., ) 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 ) or intermediary "helpers" (the ), or more general, compression followed by transmission may no longer be optimal. refers to these multi-agent communication models.

Source theory
Any process that generates successive messages can be considered a  of information. A memoryless source is one in which each message is an, whereas the properties of and  impose less restrictive constraints. All such sources are. These terms are well studied in their own right outside information theory.

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


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

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


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

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

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

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

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 $I(X; Y)$ be the distribution function of Y given X. We will consider $p(y$ to be an inherent fixed property of our communications channel (representing the nature of the  of our channel). Then the joint distribution of X and Y is completely determined by our channel and by our choice of $p(y$, 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 , we can communicate over the channel. The appropriate measure for this is the, and this maximum mutual information is called the  and is given by:
 * $$ C = \max_{f} I(X;Y).\! $$

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

 is concerned with finding such nearly optimal 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 — see.
 * A (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 $f(x)$ bits per channel use, where $1 &minus; H_{b}(p)$ is the  to the base-2 logarithm:




 * A (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 &minus; p bits per channel use.



Intelligence uses and secrecy applications
Information theoretic concepts apply to and. 's information unit, the, was used in the project, breaking the German  code and hastening the. Shannon himself defined an important concept now called the. Based on the of the, it attempts to give a minimum amount of  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 can break systems based on  or on most commonly used methods of  (sometimes called secret key algorithms), such as s.  The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time.

refers to methods such as the that are not vulnerable to such brute force attacks. In such cases, the positive conditional between the  and  (conditioned on the ) 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 was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material.

Pseudorandom number generation
s 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 s, but even they require s external to the software to work as intended. These can be obtained via, if done carefully. The measure of sufficient randomness in extractors is, a value related to Shannon entropy through ; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions among these measures mean that a 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 offer a major improvement of resolution and image clarity over previous analog methods.

Semiotics
and both considered  as having created a theory of information in his works on semiotics. Nauta defined semiotic information theory as the study of "the internal processes of coding, filtering, and information processing."

Concepts from information theory such as redundancy and code control have been used by semioticians such as and  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.