The mathematical theory of information, deriving from the work of the US mathematicians Claude E Shannon and Warren Weaver, in particular The Mathematical Theory of Communication (1949), and from the theory of probability. It is concerned with defining and measuring the amount of information in a message, with the encoding and decoding of information, and with the transmission capacity of a channel of communication. The basic notion is that the less predictable something is, the more information it contains. For example, the letter z carries a lot more information than e (in English, less so in German), when it occurs in a message. In contrast, the u after q carries no information at all, since it is wholly predictable, and thus technically redundant.
The theory also deals with the problem of noise (random interference) in the channel, which can impair the reception and decoding of a signal. To reduce the risk of errors which may arise, and thereby to increase efficiency, a signal should contain a degree of redundancy. For example, the final digit of the number identifying books (the ISBN) provides information to check that the rest of the number is valid. Information theory has been influential in generating models for understanding communication processes and in the design of codes for the transmission of information, especially by computers. It is not, however, concerned with the content, meaning, or importance of that information or any other communication.
Not to be confused with information technology, information science, or informatics.Information theory is a discipline in applied mathematics involving the quantification of data with the goal of enabling as much data as possible to be reliably stored on a medium and/or communicated over a channel. The measure of data, known as information entropy, is usually expressed by the average number of bits needed for storage or communication.
Applications of fundamental topics of information theory include ZIP files (lossless data compression), MP3s (lossy data compression), and DSL (channel coding).
Overview
The main concepts of information theory can be grasped by considering the most widespread means of human communication: language. Source coding and channel coding are the fundamental concerns of information theory. Information theory, however, does not involve message importance or meaning, as these are matters of the quality of data rather than the quantity 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 classic 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.
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. 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 deciban for a historical application.
Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition.
Historical background
See main article: History of information theory.The decisive event which established the discipline of information theory, and brought it to immediate worldwide attention, was the publication of Claude E.
Prior to this paper, limited information theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. 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 = logS = 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.
Much of the mathematics behind information theory with events of different probabilities was developed for the field of thermodynamics by Ludwig Boltzmann and J. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory.
In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that
"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."With it came the ideas of
the information entropy and redundancy of a source, and its relevance through the source coding theorem; the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem; and of course the bit—a new way of seeing the most fundamental unit of informationMathematical theory of information
See main article: Quantities of information.The mathematical theory of information 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 choice of logarithmic base in the following formulae determines the unit of information entropy that is used. The most common unit of information currently in use is the bit, based on the binary logarithm. 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 m that M could be, and p(m) = Pr(M = m), then M has
bits of entropy.
Sometimes the function H is expressed in terms of the probabilities of the distribution:
whereAn important special case of this is the binary entropy function:
The joint entropy of two discrete (not necessarily independent) random variables X and Y is merely the entropy of their pairing, (X,Y). (Note: Joint entropy should not be confused with cross entropy, despite similar notations.)
The conditional entropy of X given Y = y is the entropy X would have if it were known that Y = y.
The conditional entropy of X given random variable Y (also called the equivocation of X about Y) is the average conditional entropy over 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:
Mutual information and other information measures
One last important measure of information is the mutual information, or transinformation. This is a measure of how much information can be obtained about one random variable by observing another. This is particularly important in communication, as a sent and a received signal, although not always identical, should be able to transmit an adequate amount of information. The mutual information of X relative to Y (which represents conceptually the average amount of information about X that can be gained by observing Y) is given by:
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:
Related quantities like self-information, Pointwise Mutual Information (PMI), Kullback-Leibler divergence (information gain), and differential entropy also play a crucial role in information theory.
Channel capacity
See main article: Noisy channel coding theorem.Communications over a channel — such as an ethernet wire — is the primary motivation of information theory. How much information can one hope to communicate over a noisy (or otherwise imperfect) 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. Under these constraints, we would like to maximize the amount of information, or the signal, we can communicate over the channel. The appropriate measure for this is the transinformation, and this maximum transinformation 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 <
Channel capacity of particular model channels
A continuous-time analog communications channel subject to Gaussian noise — see Shannon–Hartley theorem. The BSC has a capacity of 1 − Hb(p) bits per channel use, where Hb is the binary entropy function: A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. The erasure represents complete loss of information about an input bit.Source theory
Any process that generates successive messages can be considered a source of information. These terms are well studied in their own right outside information theory.
Rate
Information rate is the average entropy per symbol. 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.
Applications
Coding theory
See main artilce Coding theory.
Coding theory is the most important and direct application 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 subset of Information Theory is called rate distortion theory.
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. In scenarios with more than one transmitter (the multiple-access channel), more than one receiver (the broadcast channel) or intermediary "helpers" (the relay channel), or more general networks, compression followed by transmission may no longer be optimal. Network information theory refers to these multi-agent communication models. Secrecy applications
Information theoretic concepts are widely used in making and breaking cryptographic systems.
Shannon's theory of information is extremely important in work, much more so than its use in cryptography indicates. Intelligence agencies use Information Theory to keep classified information secret, and to discover as much information as possible about an adversary, in a future-proof secure way. In general it is not possible to stop the leakage of classified information, only to slow it. Furthermore, the more people who have access to the information, and the more those people have to work with and review that information, the greater the redundancy that information acquires. It is extremely hard to contain the flow of information that has high redundancy. This inevitable leakage of classified information is due to the psychological fact that what people know does somewhat influence their behavior, however subtle that influence might be.
Pseudo Random Number generation
A good example of the application of information theory to covert signaling is the design of the Global Positioning System signal encoding.
Miscellaneous applications
Information theory also has applications in gambling and investing, black holes, and music. Hartley, "Transmission of Information," Bell System Technical Journal, July 1928 J. Kelly, Jr., "A New Interpretation of Information Rate," Bell System Technical Journal, Vol. Landauer, "Information is Physical" Proc. 3, 1961
Textbooks on information theory
Claude E. Elements of information theory, 1st Edition. An Introduction to Information Theory. A First Course in Information Theory Kluwer Academic/Plenum Publishers, 2002. Introduction to Information Theory. ISBN 0-14-006748-5 Leon Brillouin, Science and Information Theory, Mineola, N.Y.: Dover, [1956, 1962] 2004. Khinchin, Mathematical Foundations of Information Theory, New York: Dover, 1957. Rex, Editors, Maxwell's Demon: Entropy, Information, Computing, Princeton University Press, Princeton, NJ (1990). ISBN 0-670-03441-XApplications
Cryptography Cryptanalysis Entropy in thermodynamics and information theory Intelligence (information gathering) GamblingHistory
History of information theory Timeline of information theory Shannon, C.E.Theory
| Coding theory Source coding Detection theory Estimation theory Fisher information Kolmogorov complexity Information Algebra | Information geometry Information theory and measure theory Logic of information Network coding Quantum information science Semiotic information theory Philosophy of Information |
Concepts
| Self-information Information entropy Joint entropy Conditional entropy Redundancy Channel (communications) Communication source Receiver (information theory) Rényi entropy | Mutual information Pointwise Mutual Information (PMI) Differential entropy Kullback-Leibler divergence Channel capacity Unicity distance Covert channel Encoder Decoder |
User Comments Add a comment…