Claude Shannon

In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Claude Elwood Shannon (April 30, 1916 _ February 24, 2001) has been called the father of information theory, and was the founder of practical digital circuit design theory. ... A bundle of optical fiber. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

Entropy quantifies information in a piece of data. ASCII characters chosen uniformly at random have an entropy of exactly 7 bits per character, since knowing the previous characters does not help in predicting future characters. A long string of As has an entropy of 0, since every character is predictable. The entropy of English text is between 1.0 and 1.5 bits per letter^[1]. Entropy is also the shortest average message length, in bits, that can be sent to communicate the true value of the random variable to a recipient. This represents a fundamental mathematical limit on the best possible lossless data compression of any communication: the shortest average number of bits that can be sent to communicate one message out of all the possibilities is the Shannon entropy. This article is about the unit of information. ... In computer science and information theory, data compression or source coding is the process of encoding information using fewer bits (or other information-bearing units) than an unencoded representation would use through use of specific encoding schemes. ...

Equivalently, the Shannon entropy is a measure of the average information content the recipient is missing when they do not know the value of the random variable. This article is in need of attention. ...

The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". Claude Elwood Shannon (April 30, 1916 - February 24, 2001) has been called the father of information theory, and was the founder of practical digital circuit design theory. ... A Mathematical Theory of Communication, published in 1948 by mathematician and computer scientist Claude Shannon, was one of the founding works of the field of information theory. ...

Definition

The information entropy of a discrete random variable X, that can take on possible values {x₁...x_n} is In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ...

where

I(X) is the information content or self-information of X, which is itself a random variable; and

p(x_i) = Pr(X=x_i) is the probability mass function of X.

In information theory, self-information is measure of the information content associated with the outcome of a random variable. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...

Characterisation

Information entropy is characterised by these desiderata: In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first...

(Define and )

Continuity

The measure should be continuous — i.e., changing the value of one of the probabilities by a very small amount should only change the entropy by a small amount.

Symmetry

The measure should be unchanged if the outcomes x_i are re-ordered.

etc.

Maximum

If all the outcomes are equally likely, then entropy should be maximal. In this case, the entropy increases with the number of outcomes.

Additivity

The amount of entropy should be the same independently of how the process is regarded as being divided into parts.

This last functional relationship characterizes the entropy of a system with sub-systems. It demands that the entropy of a system can be calculated from the entropy of its sub-systems if we know how the sub-systems interact with each other.

Assume that we have an ensemble of n elements with a uniform distribution on them. If we mentally divide this ensemble into k boxes (sub-systems) with b_i elements in each, the entropy can be calculated as a sum of individual entropies of the boxes weighed by the probability of finding oneself in that particular box PLUS the entropy of the system of boxes.

For positive integers b_i where b₁ + … + b_k = n,

This implies that the entropy of a certain outcome is zero:

Any definition of entropy satisfying these assumptions has the form: In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

where K is a constant corresponding to a choice of measurement units.

Example

Entropy of a Coin toss as a function of the probability of it coming up heads

Main article: binary entropy function

Consider tossing a coin which may or may not be fair. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Entropy of a Bernoulli trial as a function of success probability, called the binary entropy function. ...

The entropy of the unknown result of the next toss of the coin is maximised if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers a full 1 bit of information. This article is about the unit of information. ...

However, if we know the coin is not fair, there is less uncertainty. Every time, one side is more likely to come up than the other. The reduced uncertainty is quantified in a lower entropy: on average each toss of the coin delivers less than a full 1 bit of information.

The extreme case is that of a double-headed coin which never comes up tails. Then there is no uncertainty. The entropy is zero: each toss of the coin delivers no information.

Further properties

The Shannon entropy satisfies the following properties:

• Adding or removing an event with probability zero does not contribute to the entropy:

.

• It can be confirmed using the Jensen inequality that

.

In mathematics, Jensens inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. ...

Aspects

Relationship to thermodynamic entropy

Main article: Entropy in thermodynamics and information theory

The inspiration for adopting the word entropy in information theory came from the close resemblance between Shannon's formula and very similar known formulae from thermodynamics. This article or section is in need of attention from an expert on the subject. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

In statistical thermodynamics the most general formula for the thermodynamic entropy S of a thermodynamic system is the Gibbs entropy, Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... For other uses of the term entropy, see Entropy (disambiguation) The thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermodynamics, specifically in statistical mechanics, the Gibbs entropy is the usual statistical mechanical entropy of a thermodynamic system, where the summation is taken over the possible states of the system as a whole (typically a 6N-dimensional space, if the system contains N separate particles). ...

defined by J. Willard Gibbs in 1878 after earlier work by Boltzmann (1872). Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American physical chemist. ... Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... Year 1872 (MDCCCLXXII) was a leap year starting on Monday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Saturday of the 12-day slower Julian calendar). ...

The Gibbs entropy translates over almost unchanged into the world of quantum physics to give the von Neumann entropy, introduced by John von Neumann in 1927, Fig. ... Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ... John von Neumann (Hungarian Margittai Neumann János Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician and polymath who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical... 1927 (MCMXXVII) was a common year starting on Saturday (link will display full calendar). ...

where ρ is the density matrix of the quantum mechanical system. A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ...

At an everyday practical level the links between information entropy and thermodynamic entropy are not close. Physicists and chemists are apt to be more interested in changes in entropy as a system spontaneously evolves away from its initial conditions, in accordance with the second law of thermodynamics, rather than an unchanging probability distribution. And, as the numerical smallness of Boltzmann's constant k_B indicates, the changes in S/k_B for even minute amounts of substances in chemical and physical processes represent amounts of entropy which are large right off the scale compared to anything seen in data compression or signal processing. The second law of thermodynamics is an expression of the universal law of increasing entropy. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

But, at a more philosophical level, connections can be made between thermodynamic and informational entropy, although it took many years in the development of the theories of statistical mechanics and information theory to make the relationship fully apparent. In fact, in the view of Jaynes (1957), thermodynamics should be seen as an application of Shannon's information theory: the thermodynamic entropy is interpreted as being an estimate of the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains uncommunicated by a description solely in terms of the macroscopic variables of classical thermodynamics. For example, adding heat to a system increases its thermodynamic entropy because it increases the number of possible microscopic states that it could be in, thus making any complete state description longer. (See article: maximum entropy thermodynamics). Maxwell's demon (hypothetically) reduces the thermodynamic entropy of a system using information about the states of individual molecules; but, as Landauer (from 1961) and co-workers have shown, the demon himself must increase his own thermodynamic entropy in the process, by at least the amount of Shannon information he proposes to first acquire and store; and so the total entropy does not decrease (which resolves the paradox). Edwin Thompson Jaynes (July 5, 1922 – April 30, 1998) was Wayman Crow Distinguished Professor of Physics at Washington University in St. ... 1957 (MCMLVII) was a common year starting on Tuesday of the Gregorian calendar. ... In physics the Maximum entropy school of thermodynamics (or more colloquially, the MaxEnt school of thermodynamics), initiated with two papers published in the Physical Review by Edwin T. Jaynes in 1957, views statistical mechanics as an inference process: a specific application of inference techniques rooted in information theory, which relate... Maxwells demon is an 1867 thought experiment by the Scottish physicist James Clerk Maxwell, meant to raise questions about the possibility of violating the second law of thermodynamics. ... Rolf Landauer (1927 – 1999) was an IBM physicist who in 1961 demonstrated that when information is lost in an irreversible circuit, the information becomes entropy and an associated amount of energy is dissipated as heat. ... 1961 (MCMLXI) was a common year starting on Sunday (the link is to a full 1961 calendar). ...

Entropy as information content

Main article: Shannon's source coding theorem

Entropy is defined in the context of a probabilistic model. Independent fair coin flips have an entropy of 1 bit per flip. A source that always generates a long string of A's has an entropy of 0, since the next character will always be an 'A'. In information theory, Shannons source coding theorem (or noiseless coding theorem) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy. ...

The entropy rate of a data source means the average number of bits per symbol needed to encode it. Empirically, it seems that entropy of English text is between .6 and 1.3 bits per character, though clearly that will vary from one source of text to another. Shannon's experiments with human predictors show an information rate of between .6 and 1.3 bits per character, depending on the experimental setup; the PPM compression algorithm can achieve a compression ratio of 1.5 bits per character. This article is about the unit of information. ... PPM is an adaptive statistical data compression technique based on context modeling and prediction. ...

From the preceding example, note the following points:

1. The amount of entropy is not always an integer number of bits.
2. Many data bits may not convey information. For example, data structures often store information redundantly, or have identical sections regardless of the information in the data structure.

Shannon's definition of entropy, when applied to an information source, can determine the minimum channel capacity required to reliably transmit the source as encoded binary digits. The formula can be derived by calculating the mathematical expectation of the amount of information contained in a digit from the information source. See also Shannon-Hartley theorem. In information theory, the Shannon-Hartley theorem states the maximum amount of error-free digital data (that is, information) that can be transmitted over a communication link with a specified bandwidth in the presence of noise interference. ...

Shannon's entropy measures the information contained in a message as opposed to the portion of the message that is determined (or predictable). Examples of the latter include redundancy in language structure or statistical properties relating to the occurrence frequencies of letter or word pairs, triplets etc. See Markov chain. In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. ...

Data compression

Entropy effectively bounds the performance of the strongest lossless (or nearly lossless) compression possible, which can be realized in theory by using the typical set or in practice using Huffman, Lempel-Ziv or arithmetic coding. The performance of existing data compression algorithms is often used as a rough estimate of the entropy of a block of data. In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. ... In computer science and information theory, Huffman coding is an entropy encoding algorithm used for lossless data compression. ... LZW (Lempel-Ziv-Welch) is an implementation of a lossless data compression algorithm created by Abraham Lempel and Jacob Ziv. ... Arithmetic coding is a method for lossless data compression. ...

Limitations of entropy as information content

Although entropy is often used as a characterization of the information content of a data source, this information content is not absolute: it depends crucially on the probabilistic model. A source that always generates the same symbol has an entropy of 0, but the definition of what a symbol is depends on the alphabet. Consider a source that produces the string ABABABABAB... in which A is always followed by B and vice versa. If the probabilistic model considers individual letters as independent, the entropy rate of the sequence is 1 bit per character. But if the sequence is considered as "AB AB AB AB AB..." with symbols as two-character blocks, then the entropy rate is 0 bits per character.

However, if we use very large blocks, then the estimate of per-character entropy rate may become artificially low. This is because in reality, the probability distribution of the sequence is not knowable exactly; it is only an estimate. For example, suppose one considers the text of every book ever published as a sequence, with each symbol being the text of a complete book. If there are N published books, and each book is only published once, the estimate of the probability of each book is 1/N, and the entropy (in bits) is -log₂ N. As a practical code, this corresponds to assigning each book a unique identifier and using it in place of the text of the book whenever one wants to refer to the book. This is enormously useful for talking about books, but it is not so useful for characterizing the information content of an individual book, or of language in general: it is not possible to reconstruct the book from its identifier without knowing the probability distribution, that is, the complete text of all the books. The key idea is that the complexity of the probabilistic model must be considered. Kolmogorov complexity is a theoretical generalization of this idea that allows the consideration of the information content of a sequence independent of any particular probability model; it considers the shortest program for a universal computer that outputs the sequence. A code that achieves the entropy rate of a sequence for a given model, plus the codebook (i.e. the probabilistic model), is one such program, but it may not be the shortest. The International Standard Book Number, or ISBN (sometimes pronounced is-ben), is a unique[1] identifier for books, intended to be used commercially. ... In computer science, the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic entropy, or program-size complexity) of an object such as a piece of text is a measure of the computational resources needed to specify the object. ... A computer program is a collection of instructions that describe a task, or set of tasks, to be carried out by a computer. ... The Turing machine is an abstract machine introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or mechanical procedure. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. ...

Data as a Markov process

A common way to define entropy for text is based on the Markov model of text. For an order-0 source (each character is selected independent of the last characters), the binary entropy is: In mathematics, a (discrete-time) Markov chain is a discrete-time stochastic process with the Markov property. ...

where p_i is the probability of i. For a first-order Markov source (one in which the probability of selecting a character is dependent only on the immediately preceding character), the entropy rate is: In mathematics, a Markov chain, named after Andrey Markov, is a discrete-time stochastic process with the Markov property. ... The entropy rate of a stochastic process is, informally, the time density of the average information in a stochastic process. ...

where i is a state (certain preceding characters) and p_i(j) is the probability of j given i as the previous character (s).

For a second order Markov source, the entropy rate is

In general the b-ary entropy of a source = (S,P) with source alphabet S = {a₁, …, a_n} and discrete probability distribution P = {p₁, …, p_n} where p_i is the probability of a_i (say p_i = p(a_i)) is defined by: In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ...

Note: the b in "b-ary entropy" is the number of different symbols of the "ideal alphabet" which is being used as the standard yardstick to measure source alphabets. In information theory, two symbols are necessary and sufficient for an alphabet to be able to encode information, therefore the default is to let b = 2 ("binary entropy"). Thus, the entropy of the source alphabet, with its given empiric probability distribution, is a number equal to the number (possibly fractional) of symbols of the "ideal alphabet", with an optimal probability distribution, necessary to encode for each symbol of the source alphabet. Also note that "optimal probability distribution" here means a uniform distribution: a source alphabet with n symbols has the highest possible entropy (for an alphabet with n symbols) when the probability distribution of the alphabet is uniform. This optimal entropy turns out to be . In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... In mathematics, the uniform distributions are simple probability distributions. ...

Efficiency

A source alphabet encountered in practice should be found to have a probability distribution which is less than optimal. If the source alphabet has n symbols, then it can be compared to an "optimized alphabet" with n symbols, whose probability distribution is uniform. The ratio of the entropy of the source alphabet with the entropy of its optimized version is the efficiency of the source alphabet, which can be expressed as a percentage. The percent sign. ...

This implies that the efficiency of a source alphabet with n symbols can be defined simply as being equal to its n-ary entropy. See also Redundancy (information theory). Redundancy in information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message. ...

Extending discrete entropy to the continuous case: differential entropy

The Shannon entropy is restricted to finite sets. The formula

where f denotes a probability density function on the real line, is analogous to the Shannon entropy and could thus be viewed as an extension of the Shannon entropy to the domain of real numbers. Formula (*) is usually referred to as the continuous entropy, or differential entropy. Although the analogy between both functions is suggestive, the following question must be set: is the Boltzmann entropy a valid extension of the Shannon entropy? To answer this question, we must establish a connection between the two functions: In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... Differential entropy (also referred to as continuous entropy) is a concept in information theory which tries to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions. ...

We wish to obtain a generally finite measure as the bin size goes to zero. In the discrete case, the bin size is the (implicit) width of each of the n (finite or infinite) bins whose probabilities are denoted by p_n. As we generalize to the continuous domain, we must make this width explicit. In statistics, a histogram is a graphical display of tabulated frequencies. ...

To do this, start with a continuous function f discretized as shown in the figure. As the figure indicates, by the mean-value theorem there exists a value x_i in each bin such that

and thus the integral of the function f can be approximated (in the Riemannian sense) by

where this limit and bin size goes to zero are equivalent.

We will denote

and expanding the logarithm, we have

As , we have

and so

But note that as , therefore we need a special definition of the differential or continuous entropy:

which is, as said before, referred to as the differential entropy. This means that the differential entropy is not a limit of the Shannon entropy for n → ∞

It turns out as a result that, unlike the Shannon entropy, the differential entropy is not in general a good measure of uncertainty or information. For example, the differential entropy can be negative; also it is not invariant under continuous co-ordinate transformations.

More useful for the continuous case is the relative entropy of a distribution, defined as the Kullback-Leibler divergence from the distribution to a reference measure m(x), In probability theory and information theory, the Kullback-Leibler divergence (or information divergence, or information gain, or relative entropy) is a natural distance measure from a true probability distribution P to an arbitrary probability distribution Q. Typically P represents data, observations, or a precise calculated probability distribution. ...

The relative entropy carries over directly from discrete to continuous distributions, and is invariant under co-ordinate reparametrisations.

References

1. ^ Schneier, B: Applied Cryptography, Second edition, page 234. John Wiley and Sons.

This article incorporates material from Shannon's entropy on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

See also

• Binary entropy function - the entropy of a Bernoulli trial with probability of success p
• Conditional entropy
• Entropy rate
• Cross entropy – is a measure of the average number of bits needed to identify an event from a set of possibilities between two probability distributions
• Entropy encoding - a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols.
• Hamming distance
• Joint entropy - is the measure how much entropy is contained in a joint system of two random variables.
• Kolmogorov-Sinai entropy in dynamical systems
• Levenshtein distance
• Perplexity
• Quantum relative entropy - a measure of distinguishability between two quantum states.
• Rényi entropy - a generalisation of information entropy; it is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system.
• Theil index

Entropy of a Bernoulli trial as a function of success probability, called the binary entropy function. ... In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ... The conditional entropy is an entropy measure used in information theory. ... The entropy rate of a stochastic process is, informally, the time density of the average information in a stochastic process. ... In information theory, the cross entropy between two probability distributions measures the overall difference between the two distributions. ... An entropy encoding is a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols. ... In information theory, the Hamming distance, named after Richard Hamming, is the number of positions in two strings of equal length for which the corresponding elements are different. ... The joint entropy is an entropy measure used in information theory. ... In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ... The Lorenz attractor is an example of a non-linear dynamical system. ... In information theory and computer science, the Levenshtein distance is a string metric which is one way to measure edit distance. ... Perplexity is a measurement in information theory. ... In quantum information theory, quantum relative entropy is a measure of distinguishability between two quantum states. ... In information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. ... The Theil index, derived by econometrician Henri Theil, is a statistic used to measure economic inequality. ...

External links

• Information is not entropy, information is not uncertainty ! - a discussion of the use of the terms "information" and "entropy".
• I'm Confused: How Could Information Equal Entropy? - a similar discussion on the bionet.info-theory FAQ.
• Description of information entropy from "Tools for Thought" by Howard Rheingold
• A java applet representing Shannon's Experiment to Calculate the Entropy of English