Information theory

Information theory is the scientific study of the quantification, storage, and communication of digital

information

. The field was fundamentally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory,

statistics

, computer science, statistical mechanics, information engineering, and

electrical engineering

.

A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a

die

(with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory and information-theoretic security.

Applications of fundamental topics of information theory include source coding/data compression (e.g. for ZIP files), and channel coding/error detection and correction (e.g. for DSL). Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the

compact disc

, the feasibility of mobile phones and the development of the Internet. The theory has also found applications in other areas, including statistical inference,

cryptography

, neurobiology,

perception

, linguistics, the evolution and function of molecular codes (

bioinformatics

), thermal physics, molecular dynamics, quantum computing,

black holes

, information retrieval, intelligence gathering, plagiarism detection, pattern recognition, anomaly detection and even art creation.

Overview

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

Coding theory is concerned with finding explicit methods, called ''codes'', for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity. These codes can be roughly subdivided into data compression (source coding) and error-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible.

A third class of information theory codes are cryptographic algorithms (both codes and

cipher

s). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. ''See the article ban (unit) for a historical application.''

Historical background

The landmark event ''establishing'' the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the ''Bell System Technical Journal'' in July and October 1948.

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 (recalling Boltzmann's constant), where ''W'' is the speed of transmission of intelligence, ''m'' is the number of different voltage levels to choose from at each time step, and ''K'' is a constant. Ralph Hartley's 1928 paper, ''Transmission of Information'', uses the word ''information'' as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as , where ''S'' was the number of possible symbols, and ''n'' the number of symbols in a transmission. The unit of information was therefore the

decimal digit

, which since has sometimes been called the hartley in his honor as a unit or scale or measure of information.

Alan Turing

in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.

Much of the mathematics behind information theory with events of different probabilities were developed for the field of

thermodynamics

by Ludwig Boltzmann and J. Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in ''Entropy in thermodynamics and information theory''.

In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion:
:"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."

With it came the ideas of
* the information entropy and redundancy of a source, and its relevance through the source coding theorem;
* the mutual information, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem;
* the practical result of the Shannon–Hartley law for the channel capacity of a Gaussian channel; as well as
* the bit—a new way of seeing the most fundamental unit of information.

Quantities of information

Information theory is based on probability theory and statistics. Information theory often concerns itself with measures of information of the distributions associated with random variables. Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, 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 compressed. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution.

The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm.

In what follows, an expression of the form is considered by convention to be equal to zero whenever . This is justified because $\lim_ p \log p = 0$ for any logarithmic base.

Entropy of an information source

Based on the probability mass function of each source symbol to be communicated, the Shannon entropy , in units of bits (per symbol), is given by
:$H = - \sum_ p_i \log_2 (p_i)$
where is the probability of occurrence of the -th possible value of the source symbol. This equation gives the entropy in the units of "bits" (per symbol) because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called the shannon in his honor. Entropy is also commonly computed using the natural logarithm (base , where is Euler's number), which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas. Other bases are also possible, but less commonly used. For example, a logarithm of base will produce a measurement in bytes per symbol, and a logarithm of base 10 will produce a measurement in decimal digits (or hartleys) per symbol.

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

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

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 $\mathbb$ is the set of all messages that could be, and is the probability of some $x \in \mathbb X$, then the entropy, , of is defined:

: H(X) = \mathbb_ (x)= -\sum_ p(x) \log p(x).

(Here, is the self-information, which is the entropy contribution of an individual message, and $\mathbb_X$ is the expected value.) A property of entropy is that it is maximized when all the messages in the message space are equiprobable ; i.e., most unpredictable, in which case .

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

:$H_(p) = - p \log_2 p - (1-p)\log_2 (1-p).$

Joint entropy

The of two discrete random variables and is merely the entropy of their pairing: . This implies that if and are independent, then their joint entropy is the sum of their individual entropies.

For example, if represents the position of a chess piece— the row and 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_ \log p(x,y)= - \sum_ 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 given random variable (also called the ''equivocation'' of about ) is the average conditional entropy over :

: H(X, Y) = \mathbb E_Y y)= -\sum_ p(y) \sum_ p(x, y) \log p(x, y) = -\sum_ 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)

''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 relative to is given by:

:I(X;Y) = \mathbb_ I(x,y)= \sum_ p(x,y) \log \frac
where (''S''pecific mutual ''I''nformation) is the pointwise mutual information.

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 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) between the posterior probability distribution of ''X'' given the value of ''Y'' and the prior distribution on ''X'':
: I(X;Y) = \mathbb E_ 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:
: $I(X; Y) = D_(p(X,Y) \, p(X)p(Y)).$

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

Kullback–Leibler divergence (information gain)

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

:$D_(p(X) \, q(X)) = \sum_ -p(x) \log \, - \, \sum_ -p(x) \log = \sum_ p(x) \log \frac.$

Although it is sometimes used as a 'distance metric', KL divergence is not a true metric since it is not symmetric and does not satisfy the

triangle inequality

(making it a semi-quasimetric).

Another interpretation of the KL divergence is the "unnecessary surprise" introduced by a prior from the truth: suppose a number ''X'' is about to be drawn randomly from a discrete set with probability distribution . If Alice knows the true distribution , while Bob believes (has a prior

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