Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E. Shannon to find fundamental limits on compressing and reliably storing and communicating data. Since its inception it has broadened to find applications in many other areas, including statistical inference, natural language processing, cryptography generally, networks other than communication networks — as in neurobiology,^{[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.^{[7]}
A key measure of information in the theory is known as entropy, which is usually expressed by the average number of bits needed for storage or communication. Intuitively, entropy quantifies the uncertainty involved when encountering a random variable. For example, a fair coin flip (2 equally likely outcomes) will have less entropy than a roll of a die (6 equally likely outcomes).
Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s), and channel coding (e.g. for DSL lines). The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields. Important subfields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, and measures of information.
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The main concepts of information theory can be grasped by considering the most widespread means of human communication: language. Two important aspects of a good language are as follows: First, the most common words (e.g., "a", "the", "I") should be shorter than less common words (e.g., "benefit", "generation", "mediocre"), so that sentences will not be too long. Such a tradeoff in word length is analogous to data compression and is the essential aspect of source coding. Second, if part of a sentence is unheard or misheard due to noise — e.g., a passing car — the listener should still be able to glean the meaning of the underlying message. Such robustness is as essential for an electronic communication system as it is for a language; properly building such robustness into communications is done by channel coding. Source coding and channel coding are the fundamental concerns of information theory.
Note that these concerns have nothing to do with the importance of messages. For example, a platitude such as "Thank you; come again" takes about as long to say or write as the urgent plea, "Call an ambulance!" while clearly the latter is more important and more meaningful. Information theory, however, does not consider message importance or meaning, as these are matters of the quality of data rather than the quantity and readability of data, the latter of which is determined solely by probabilities.
Information theory is generally considered to have been founded in 1948 by Claude Shannon in his seminal work, "A Mathematical Theory of Communication." The central paradigm of classical information theory is the engineering problem of the transmission of information over a noisy channel. The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy; and Shannon's noisychannel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold called the channel capacity. The channel capacity can be approached in practice by using appropriate encoding and decoding systems.
Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half century or more: adaptive systems, anticipatory systems, artificial intelligence, complex systems, complexity science, cybernetics, informatics, machine learning, along with systems sciences of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.
Coding theory is concerned with finding explicit methods, called codes, of increasing the efficiency and reducing the net error rate of data communication over a noisy channel to near the limit that Shannon proved is the maximum possible for that channel. These codes can be roughly subdivided into data compression (source coding) and errorcorrection (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban (information) for a historical application.
Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition.
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 1948.
Prior to this paper, limited informationtheoretic 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 = Klogm, 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 = logS^{n} = nlogS, where S was the number of possible symbols, and n the number of symbols in a transmission. The natural unit of information was therefore the decimal digit, much later renamed the hartley in his honour as a unit or scale or measure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.
Much of the mathematics behind information theory with events of different probabilities was developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs. Connections between informationtheoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory.
In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that
With it came the ideas of
Information theory is based on probability theory and statistics. The most important quantities of information are entropy, the information in a random variable, and mutual information, the amount of information in common between two random variables. The former quantity indicates how easily message data can be compressed while the latter can be used to find the communication rate across a channel.
The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. The most common unit of information is the bit, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the hartley, which is based on the common logarithm.
In what follows, an expression of the form is considered by convention to be equal to zero whenever p = 0. This is justified because for any logarithmic base.
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 is the set of all messages {x_{1},...,x_{n}} that X could be, and p(x) is the probability of X given some , then the entropy of X is defined:^{[8]}
(Here, I(x) is the selfinformation, which is the entropy contribution of an individual message, and is the expected value.) An important 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) = logn.
The special case of information entropy for a random variable with two outcomes is the binary entropy function:
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.
Despite similar notation, joint entropy should not be confused with cross entropy.
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]}
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:
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:
where SI (Specific mutual Information) is the pointwise mutual information.
A basic property of the mutual information is that
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:
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:
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:
Mutual information is closely related to the loglikelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ^{2} test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a wellspecified asymptotic distribution.
The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions: a "true" probability distribution p(X), and an arbitrary probability distribution q(X). If we compress data in a manner that assumes q(X) is the distribution underlying some data, when, in reality, p(X) is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus defined
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 semiquasimetric).
Other important information theoretic quantities include Rényi entropy, (a generalization of entropy,) differential entropy, (a generalization of quantities of information to continuous distributions,) and the conditional mutual information.
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.
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 multipleaccess channel), more than one receiver (the broadcast channel) or intermediary "helpers" (the relay channel), or more general networks, compression followed by transmission may no longer be optimal. Network information theory refers to these multiagent communication models.
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 identicallydistributed random variable, whereas the properties of ergodicity and stationarity impose more general constraints. All such sources are stochastic. These terms are well studied in their own right outside information theory.
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
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
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.
Communications over a channel—such as an ethernet wire—is the primary motivation of information theory. As anyone who's ever used a telephone (mobile or landline) knows, however, such channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality. How much information can one hope to communicate over a noisy (or otherwise imperfect) channel?
Consider the communications process over a discrete channel. A simple model of the process is shown below:
Here X represents the space of messages transmitted, and Y the space of messages received during a unit time over our channel. Let p(y  x) be the conditional probability distribution function of Y given X. We will consider p(y  x) to be an inherent fixed property of our communications channel (representing the nature of the noise of our channel). Then the joint distribution of X and Y is completely determined by our channel and by our choice of f(x), the marginal distribution of messages we choose to send over the channel. Under these constraints, we would like to maximize the rate of information, or the signal, we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by:
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.
Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of WWII in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.
Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms (sometimes called secret key algorithms), such as block ciphers. The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time.
Information theoretic security refers to methods such as the onetime 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 informationtheoretically secure methods; the Venona project was able to crack the onetime pads of the Soviet Union due to their improper reuse of key material.
Pseudorandom number generators are widely available in computer language libraries and application programs. They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software. A class of improved random number generators is termed Cryptographically secure pseudorandom number generators, but even they require external to the software random seeds to work as intended. These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractors is minentropy, 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.
One early commercial application of information theory was in the field 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.^{[11]}
Information theory also has applications in gambling and investing, black holes, bioinformatics, and music.


Information theory is a branch of mathematics dealing with the representation, storage, and transmission of information. It has found important applications in biology, electrical engineering, linguistics, and computer science.
This learning resource is intended to give an intuitive introduction to the ideas of information theory. We encourage you to follow the links above and below to other online resources to broaden the understanding you may begin to build here.
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Have you ever played the parlor game called Twenty Questions? It's both easy and fun: one person (Answerer) thinks of some specific object (for example, the Earth), and the other (Questioner) asks questions to figure out which object Answerer has chosen. Questioner can ask any question about the object, so long as it can be answered simply "yes" or "no". One question to ask first might be "Is it alive?" Since the Earth itself isn't alive, Answerer says "no" (Answerer must tell the truth). This answered question gives Questioner information about what the object is and helps Questioner decide which question to ask next. If Questioner guesses the object by the 20^{th} question, Questioner wins; otherwise Answerer wins.
Information theory is simply a way to think about quantifying, representing, and communicating information of the sort the answered question above provides. To be more precise, information theory is relevant to any setting in which there is a set of events {x_{i}} along with one or more ideas of how likely each event is to occur (or to have occurred). A particular idea of how likely each event is to occur is a probability function P = {p_{i}}, where is the probability that event x_{i} will occur. In Twenty Questions, the events refer to Answerer's choice of an object: "Answerer chose a tiger", "Answerer chose the Earth", or "Answerer chose his left thumb" are all events that might have occurred at the start of the game. In general we can think of event x_{i} as referring to "Answerer chose the i^{th} object from a master list of all objects that exist.
At the start of a game of Twenty Questions, Questioner doesn't know anything about the object Answerer has chosen. That is, as far as Questioner knows any object is equally likely to be the one Answerer has thought of: a bacterium, Answerer's left thumb, the entire universe. In mathematical terms, p_{i} = p_{j} for any two events i and j. After the first question is answered, Answerer forms a new probability function from the new information. In information theory we assume that Questioner understands the full importance of the information the answered question provided, because the information is there to be used whether Questioner is a perfect player or not. Most of the questions asked in Twenty Questions will rule out some set of objects entirely, however in other settings new information will change the probability function in other ways, making some events seem more likely or others seem less likely. It is the difference between the best probability function that could have been formed before a question was asked, taking full advantage of everything known to that point, and the best probability function available after the question was answered which illustrates how much information the answered question provided.
Quantities of information are relevant to a lot of different questions, such as:
The standard unit of information is the bit: one bit is the amount of information provided by choosing one of two equally likely possibilities. One toss of a fair coin provides one bit of information: before tossing the coin either heads or tails is equally likely to be the result, but afterwards we know whether the coin landed heads up or tails up.
We can now answer the first question above: how many yes/no questions does it take to guess a number between 1 and 3 billion? The very best we can do is to ask if the number is in the upper half or lower half of the range, until the range is just a single number. Doing so eliminates half the possibilities on each question, no matter whether the answer is "upper half" or "lower half". The number of questions is therefore the number of times one can divide the total range in half before getting to a range of just one number; i.e. what is the smallest N such that 2^{N} is at least 3 billion? If you were served exactly one of the first 3 billion hamburgers served by McDonald's, knowing exactly which one you were served gives you about 32 bits of information, because 3,000,000,000 must be halved 32 times to reach a range of only 1 number. In general, guessing a number between 1 and N requires log_{2}(N) bits of information.
TBD
TBD
Let {x_{i}} be a finite set of possible events (for example, possible event i corresponds to Answerer chose the i^{th} object from a known list of all objects Answerer is allowed to choose in a game of Twenty Questions). We can represent the full information we have about which of these possibilities is the truth (which object Answerer chose) as a discrete probability distribution As a step toward determining the amount of information we have, we define the uncertainty or entropy of P as
H(P) = − ∑ p_{i}log_{2}(p_{i}). i
Note that if x_{i} is one of N equally likely possibilities, the amount of information needed to determine that x_{i} is the event that occurred is bits. Thus we can think of − log_{2}(p_{i}) as the amount of information required to identify that x_{i} occurred, and by weighting each x_{i} by its probability p_{i} the entropy of a probability distribution can be thought of as the average information required to identify which event occurred, under the assumption that the events occur according to the probability distribution in question (i.e. according to the information the probability distribution represents). Entropy is therefore a measure of how much doubt remains in determining which event occurred, after having taken into account the information represented by P. It is therefore appropriate to consider entropy as defined above a measure of uncertainty.
Receiving the answer to a (helpful) question reduces the amount of uncertainty we have about which event occurred. Information therefore corresponds to a reduction in uncertainty. In fact, this is how information is defined in mathematical terms. Formally, if p_{i} is the probability distribution over a finite set of possibilities before a question is asked and q_{i} is the probability distribution over the same set of possibilities after the question is asked, then the information provided by the answered question is
Information
The information provided by a question that changes the probability of each object from p to q is therefore defined as the reduction in entropy from p to q.
Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd Ed., WileyInterscience, 2006. A modern textbook.
Robert B. Ash, Information Theory, Dover Publications, 1990.
John R. Pierce, An Introduction to Information Theory, Dover Publications, 1980. A very readable introduction.
Claude E. Shannon and Warren Weaver, The Mathematical Theory of Communication, University of Illinois Press, 1963.
David P. Hornacek, A Brief Tutorial on Information Theory, Excess Entropy and Statistical Complexity: Discovering and Quantifying Statistical Structure, Lecture notes downloadable in several formats, 1997.
Information theory is a branch of applied mathematics and electrical engineering. Information theory measures the amount of information in data which could have more than one value. In its most common use, information theory finds physical and mathematical limits on the amounts of data in data compression and data communication. Data compression and data communication are statistical, because they guess unknown values. The amount of information in data measures how easily it is guessed by a person who does not know its value.
