Information theory

The topic of this article is distinct from the topics of Library and information science and, largely, from Information technology.

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. For example, if a daily weather description has 3 bits of entropy, then, over enough days, we can describe daily weather with an average of approximately 3 bits per day (each bit being a 0 or a 1). Applications of fundamental topics of information theory include ZIP files (lossless data compression), MP3s (lossy data compression), and DSL (channel coding). The field is at the crossroads of mathematics, statistics, computer science, physics and electrical engineering, and its impact has been crucial to success of the Voyager missions to deep space, the invention of the CD, the feasibility of mobile phones, the development of the Internet, the study of human perception, the understanding of black holes, and numerous other fields.

Overview

The main concepts of information theory can be grasped by considering the most widespread means of human communication, language. Two important aspects of a good language are as follows: First, the most common words (e.g., "a," "the," "I") should be shorter than less common words (e.g., "benefit," "generation," "mediocre"), so that sentences will not be too long. Such a tradeoff in word length is analogous to data compression and is the essential aspect of source coding. Second, if part of a sentence is unheard or misheard due to noise — e.g., a passing car — the listener should still be able to glean the meaning of the underlying message. Such robustness is as essential for an electronic communication system as it is for a language; properly building such robustness into communications is done by channel coding. 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 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. The channel capacity can be approached 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 error-correction (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 deciban for a historical application.

Information theory is also used in information retrieval, intelligence gathering, gambling, statistics, and even in musical composition.

History

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. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October of 1948.

Prior to this paper, limited information theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation $W = K \log m$ , where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish that one sequence of symbols from any other, thus quantifying information as $H = \log S^n = n \log S$ , where S was the number of possible symbols, and n the number of symbols in a transmission. 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, although Shannon himself was apparently not particularly aware of the close similarity between his new measure and earlier work in thermodynamics. (Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored further in the article Entropy in thermodynamics and information theory).

In Shannon's revolutionary and groundbreaking paper, the work for which had substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that

"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."

With it came the ideas of

the information entropy and redundancy of a source, and its relevance through the source coding theorem;
the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem;
the practical result of the Shannon-Hartley law for the channel capacity of a Gaussian channel; and of course
the bit - a new way of seeing the most fundamental unit of information

Mathematical theory of information

The mathematical theory of information is based on probability theory and statistics.

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, based on the natural logarithm, and the hartley, based on the base 10 or common logarithm. In what follows, an expression of the form $p \log p \,$ is considered by convention to be equal to zero whenever p is. This is justified because $\lim_{p \rightarrow 0+} p \log p = 0$ for any logarithmic base.

Self-information

Shannon defined a measure of information content called the self-information or surprisal of a message m:

I(m) = - \log p(m),\,

where $p(m) = Pr(M=m)$ is the probability that message m is chosen from all possible choices in the message space $M$ .

This equation weights messages with lower probabilities higher in contributing to the overall value of I(m). In other words, infrequently occurring messages are more valuable. (This is a consequence from the property of logarithms that $-\log p(m)$ is very large when $p(m)$ is near 0 for unlikely messages and very small when $p(m)$ is near 1 for almost certain messages).

For example, if John says "See you later, honey" to his wife every morning before leaving to office, that information holds little "content" or "value". But, if he shouts "Get lost" at his wife one morning, then that message holds more value or content (because, supposedly, the probability of him choosing that message is very low).

Entropy

The entropy of a discrete message space

M

is a measure of the amount of uncertainty one has about which message will be chosen. It is defined as the average self-information of a message

m

from that message space:

H(M) = \mathbb{E} \{I(m)\} = \sum_{m \in M} p(m) I(m) = -\sum_{m \in M} p(m) \log p(m).

An important property of entropy is that it is maximized when all the messages in the message space are equiprobable. In this case $H(M) = \log |M|. \,$

Sometimes the function H is expressed in terms of the probabilities of the distribution:

H(p_1, p_2, \ldots , p_k) = -\sum_{i=1}^k p_i \log p_i,

where each

p_i \geq 0

and

\sum_{i=1}^k p_i = 1.

An important special case of this is the binary entropy function:

H_\mbox{b}(p) = H(p, 1-p) = - p \log p - (1-p)\log (1-p).\,

Joint entropy

The joint entropy of two discrete random variables

X

and

Y

is defined as the entropy of the joint distribution of

X

and

Y

= - \sum_{x, y} p(x, y) \log p(x, y) \,

If $X$ and $Y$ are independent, then the joint entropy is simply the sum of their individual entropies.

(Note: The joint entropy should not be confused with the cross entropy, despite similar notations.)

Conditional entropy (equivocation)

Given a particular value of a random variable

Y

, the conditional entropy of

X

given

Y=y

is defined as:

= -\sum_{x \in X} p(x|y) \log p(x|y)

where $p(x|y) = \frac{p(x,y)}{p(y)}$ is the conditional probability of $x$ given $y$ .

The conditional entropy of $X$ given $Y$ , also called the equivocation of $X$ about $Y$ is then given by:

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

A basic property of the conditional entropy is that:

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

Kullback–Leibler divergence (information gain)

The Kullback–Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions, a "true" probability distribution p, and an arbitrary probability distribution q. If we compress data in a manner that assumes q is the distribution underlying some data, when, in reality, p is the correct distribution, Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression, or, mathematically,

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

It is in some sense the "distance" from q to p, although it is not a true metric due to its not being symmetric.

Mutual information (transinformation)

It turns out that one of the most useful and important measures 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. 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:

I(X;Y) = \sum_{y\in Y} p(y)\sum_{x\in X} p(x|y) \log \frac{p(x|y)}{p(x)} = \sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)\, p(y)}.

A basic property of the mutual information is that:

I(X;Y) = H(X) - H(X|Y).\,

That is, knowing Y, we can save an average of $I(X; Y)$ bits in encoding X compared to not knowing Y. Mutual information is symmetric:

I(X;Y) = I(Y;X) = H(X) + H(Y) - H(X,Y).\,

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

I(X;Y) = \mathbb E_{p(y)} \{D_{\mathrm{KL}}( p(X|Y=y) \| p(X) )\}.

In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y. This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution:

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

Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ² test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution.

Continuous equivalents of entropy

See main article: Differential entropy.

Shannon information is appropriate for measuring uncertainty over a discrete space. Its basic measures have been extended by analogy to continuous spaces. The sums can be replaced with integrals and densities are used in place of probability mass functions. By analogy with the discrete case, entropy, joint entropy, conditional entropy, and mutual information can be defined as follows:

h(X) = -\int_X f(x) \log f(x) \,dx

h(X,Y) = -\int_Y \int_X f(x,y) \log f(x,y) \,dx \,dy

h(X|y) = -\int_X f(x|y) \log f(x|y) \,dx

h(X|Y) = \int_Y \int_X f(x,y) \log \frac{f(y)}{f(x,y)} \,dx \,dy

I(X;Y) = \int_Y \int_X f(x,y) \log \frac{f(x,y)}{f(x)f(y)} \,dx \,dy

where $f(x,y)$ is the joint density function, $f(x)$ and $f(y)$ are the marginal distributions, and $f(x|y)$ is the conditional distribution.

Channel capacity

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 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:

C = \max_{f} I(X;Y).\!

Source theory

Any process that generates successive messages can be considered a source of information. Sources can be classified in order of increasing generality as memoryless, ergodic, stationary, and stochastic. The term "memoryless" as used here has a slightly different meaning than it normally does in probability theory. Here a memoryless source is defined as one that generates successive messages independently of one another and with a fixed probability distribution. However, the position of the first occurrence of a particular message or symbol in a sequence generated by a memoryless source is actually a memoryless random variable. The other terms have fairly standard definitions and are actually well studied in their own right outside information theory.

Rate

The rate of a source of information is (in the most general case)

r=\mathbb E H(M_t|M_{t-1},M_{t-2},M_{t-3}, \ldots),

the expected, or average, conditional entropy per message (i.e. per unit time) given all the previous messages generated. 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 memoryless source is simply $H(M)$ , since by definition there is no interdependence of the successive messages of a memoryless source. The rate of a source of information is related to its redundancy and how well it can be compressed.

Noisy channel coding theorem

See main article: Noisy channel coding theorem.

Statement

1. For every discrete memoryless channel, the channel capacity

C = \max_{P_X} \,I(X;Y)

has the following property. For any ε > 0 and R < C, 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.

2. If a probability of bit error p_b is acceptable, rates up to R(p_b) are achievable, where

R(p_b) = \frac{C}{1-H_\mbox{b}(p_b)} ,

and

H_\mbox{b}(p_b) = -p_b\log_2 p_b - (1-p_b)\log_2 (1-p_b)\,

is the binary entropy function.

3. For any p_b, rates greater than R(p_b) are not achievable.

(MacKay (2003), p. 162; cf Gallager (1968), ch.5; Cover and Thomas (1991), p. 198; Shannon (1948) thm. 11)

Channel capacity of particular model channels

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 $1 - H_\mbox{b}(p)$ bits per channel use, where $H_\mbox{b}$ is the binary entropy function:

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.

Applications

Coding theory

Coding theory is the most important and direct application of information theory. It can be subdivided into data compression theory and error correction theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data. There are two formulations for the compression problem — in lossless data compression the data must be reconstructed exactly, whereas lossy data compression examines how many bits are needed to reconstruct the data to within a specified fidelity level. This fidelity level is measured by a function called a distortion function. In information theory this is called rate distortion theory. Both lossless and lossy source codes produce bits at the output which can be used as the inputs to the channel codes mentioned above.

The idea is to first compress the data, i.e. remove as much of its redundancy as possible, and then add 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. Network information theory refers to these multi-agent communication models.

Other applications

Information theoretic concepts are widely used in making and breaking cryptographic systems. For an interesting historical example, see the article on deciban. 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.

Shannon's theory of information is extremely important in intelligence work, much more so than its use in cryptography would indicate. The theory is applied by intelligence agencies to keep classified information secret, and to discover as much information as possible about an adversary. The fundamental theorem leads us to believe it is much more difficult to keep secrets than it might first appear. 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.

A good example of the application of information theory to covert signaling is the design of the Global Positioning System signal encoding. The system uses a pseudorandom encoding that places the radio signal below the noise floor. Thus, an unsuspecting radio listener would not even be aware that there was a signal present, as it would be drowned out by aassorted noise sources (eg, atmospheric and antenna noise). However, if one integrates the signal over long periods of time, using the "secret" (but known to the listener) pseudorandom sequence, one can eventually detect a signal, and then discern modulations of that signal. In the GPS system, the C/A signal has been publicly disclosed to be a 1023-bit sequence, but the pseudorandom sequence used in the P(Y) signal remains a secret. The same technique can be used to transmit and receive covert intelligence from short-range, extremely low power systems, without an Enemy even being aware of the existence of a radio signal. This is analogous to steganography. See also spread spectrum communications.

Information theory also has applications in gambling and investing, black holes, and music.

References

The classic paper

Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27, pp. 379–423 & 623–656, July & October, 1948. PDF.
Notes and other formats.

Textbooks on information theory

Claude E. Shannon, Warren Weaver. The Mathematical Theory of Communication. Univ of Illinois Press, 1949. ISBN 0252725484
Robert Gallager. Information Theory and Reliable Communication. New York: John Wiley and Sons, 1968. ISBN 0471290483
Robert B. Ash. Information Theory. New York: Interscience, 1965. ISBN 0470034459. New York: Dover 1990. ISBN 0486665216
Thomas M. Cover, Joy A. Thomas. Elements of information theory, 1st Edition. New York: Wiley-Interscience, 1991. ISBN 0471062596.

2nd Edition. New York: Wiley-Interscience, 2006. ISBN 0471241954 (forthcoming, to be released 2006.).

Stanford Goldman. Information Theory. New York: Prentice Hall, 1953. New York: Dover 1968 ISBN 0486622096, 2005 ISBN 0486442713
Fazlollah M. Reza. An Introduction to Information Theory. New York: McGraw-Hill 1961. New York: Dover 1994. ISBN 0486682102
Raymond W. Yeung. A First Course in Information Theory Kluwer Academic/Plenum Publishers, 2002. ISBN 03-6467917
David J. C. MacKay. Information Theory, Inference, and Learning Algorithms Cambridge: Cambridge University Press, 2003. ISBN 0521642981

Other books

James Bamford, The Puzzle Palace, Penguin Books, 1983. ISBN 0140067485
Leon Brillouin, Science and Information Theory, Mineola, N.Y.: Dover, 1962 2004. ISBN 0486439186
A. I. Khinchin, Mathematical Foundations of Information Theory, New York: Dover, 1957. ISBN 0486604349
H. S. Leff and A. F. Rex, Editors, Maxwell's Demon: Entropy, Information, Computing, Princeton University Press, Princeton, NJ (1990). ISBN 069108727X
Tom Siegfried, The Bit and the Pendulum, Wiley, 2000. ISBN 0471321745
Charles Seife, Decoding The Universe, Viking, 2006. ISBN 067003441X

External links

Gibbs, M., "Quantum Information Theory", Eprint
Schneider, T., "Information Theory Primer", Eprint
Srinivasa, S. "A Review on Multivariate Mutual Information" PDF.
Journal of Chemical Education, Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? Nonsense!
IEEE Information Theory Society and the review articles.
On-line textbook: Information Theory, Inference, and Learning Algorithms, by David MacKay - gives an entertaining and thorough introduction to Shannon theory, including state-of-the-art methods from coding theory, such as arithmetic coding, low-density parity-check codes, and Turbo codes.
Blog article on Information Theory

Communication | Cybernetics | Discrete mathematics | Information theory

Gourt Home::Gourt :: Information theory

Overview

History

Mathematical theory of information

Self-information

Entropy

Joint entropy

Conditional entropy (equivocation)

Kullback–Leibler divergence (information gain)

Mutual information (transinformation)

Continuous equivalents of entropy

Channel capacity

Source theory

Rate

Noisy channel coding theorem

Statement

Channel capacity of particular model channels

Applications

Coding theory

Other applications

References

The classic paper

Other journal articles

Textbooks on information theory

Other books

See also

Applications

History

Theory

Concepts

External links