# Information theory

Information theory |
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**Information theory** is the mathematical study of the quantification, storage, and communication of information. The field was established and put on a firm footing by Claude Shannon in the 1940s,^{[1]} though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, neurobiology, physics, and electrical engineering.^{[2]}^{[3]}

A key measure in information theory is

Applications of fundamental topics of information theory include source coding/

^{[18]}study of outer space,

^{[19]}the dimensionality of space,

^{[20]}and epistemology.

^{[21]}

## 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.^{[7]}

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

^{[citation needed}

A third class of information theory codes are

^{[citation needed}

## 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. Historian James Gleick rated the paper as the most important development of 1948, above the transistor, noting that the paper was "even more profound and more fundamental" than the transistor.

^{[22]}He came to be known as the "father of information theory".

^{[23]}

^{[24]}

^{[25]}Shannon outlined some of his initial ideas of information theory as early as 1939 in a letter to Vannevar Bush.

^{[25]}

Prior to this paper, limited information-theoretic ideas had been developed at

^{[citation needed}

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

*Entropy in thermodynamics and information theory*.

^{[citation needed}

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 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 Gaussian channel; as well as
- the bit—a new way of seeing the most fundamental unit of information.
^{[citation needed]}

## Quantities of information

Information theory is based on probability theory and statistics, where quantified information is usually described in terms of bits. Information theory often concerns itself with measures of information of the distributions associated with random variables. One of the most important measures is called entropy, which forms the building block of many other measures. Entropy allows quantification of measure of information in a single random variable.^{[26]} Another useful concept is mutual information defined on two random variables, which describes the measure of information in common between those variables, which can be used to describe their correlation. 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

In what follows, an expression of the form *p* log *p* is considered by convention to be equal to zero whenever *p* = 0. This is justified because 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 *H*, in units of bits (per symbol), is given by

where *p _{i}* is the probability of occurrence of the

*i*-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 e, where e 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 2

^{8}= 256 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 *H _{X}* of a discrete random variable

*X*is a measure of the amount of

*uncertainty*associated with the value of

*X*when only its distribution is known.

The entropy of a source that emits a sequence of *N* symbols that are

*N*⋅

*H*bits (per message of

*N*symbols). If the source data symbols are identically distributed but not independent, the entropy of a message of length

*N*will be less than

*N*⋅

*H*.

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 is the set of all messages {*x*_{1}, ..., *x*_{n}} that *X* could be, and *p*(*x*) is the probability of some , then the entropy, *H*, of *X* is defined:^{[27]}

(Here, *I*(*x*) is the

*p*(

*x*) = 1/

*n*; i.e., most unpredictable, in which case

*H*(

*X*) = log

*n*.

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:

### Joint entropy

The

*independent*

*, then their joint entropy is the sum of their individual entropies.
metric since it is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric).
pragmatic information has been proposed as a measure of how much information has been used in making a decision.
*

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*.

### Conditional entropy (equivocation)

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*:^{[28]}

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 (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 *X* relative to *Y* is given by:

where SI (*S*pecific 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) between the posterior probability distribution of *X* given the value of *Y* and 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 log-likelihood 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 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

Although it is sometimes used as a 'distance metric', KL divergence is not a true

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) that the distribution is , then Bob will be more surprised than Alice, on average, upon seeing the value of *X*. The KL divergence is the (objective) expected value of Bob's (subjective) surprisal minus Alice's surprisal, measured in bits if the *log* is in base 2. In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him.

### Directed Information

**Directed information**, , is an information theory measure that quantifies the information flow from the random process to the random process . The term *directed information* was coined by James Massey and is defined as

- ,

where is the conditional mutual information .

In contrast to *mutual* information, *directed* information is not symmetric. The measures the information bits that are transmitted causally^{[clarification needed]} from to . The Directed information has many applications in problems where

### Other quantities

Other important information theoretic quantities include the

## Coding theory

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.

- Data compression (source coding): There are two formulations for the compression problem:
- lossless data compression: the data must be reconstructed exactly;
- lossy data compression: allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. This subset of information theory is called.
*rate–distortion theory*

- Error-correcting codes(channel coding): While data compression removes as much redundancy as possible, an error-correcting code adds 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

, compression followed by transmission may no longer be optimal.### Source theory

Any process that generates successive messages can be considered a

*independent identically distributed random variable*

*, whereas the properties of ergodicity and stationarity impose less restrictive constraints. All such sources are stochastic*

*. These terms are well studied in their own right outside information theory.
information rate is defined as:
signal, we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the *

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.
encoding is not important.

ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II 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.
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 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 one-time 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 information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material.

Bayesian brain hypothesis).
#### Rate

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.^{[37]}

The

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*.

### Channel capacity

Communications over a channel is the primary motivation of information theory. However, channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality.

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

*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*

#### Capacity of particular channel models

- 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*_{b}(*p*) bits per channel use, where*H*_{b}is the binary entropy function to the base-2 logarithm:

- 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.

#### Channels with memory and directed information

In practice many channels have memory. Namely, at time the channel is given by the conditional probability.
It is often more comfortable to use the notation and the channel become .
In such a case the capacity is given by the mutual information rate when there is no feedback available and the Directed information rate in the case that either there is feedback or not^{[29]}^{[38]} (if there is no feedback the directed information equals the mutual information).

### Fungible information

**Fungible information** is the

^{[39]}Classical information theorists and computer scientists are mainly concerned with information of this sort. It is sometimes referred as speakable information.

^{[40]}

## Applications to other fields

### Intelligence uses and secrecy applications

Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the

Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A

### Pseudorandom number generation

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 random seeds external to the software to work as intended. These can be obtained via extractors, if done carefully. The measure of sufficient randomness in extractors is min-entropy, 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.

### Seismic exploration

One early commercial application of information theory was in the field of 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.^{[41]}

### Semiotics

Semioticians Doede Nauta and Winfried Nöth both considered Charles Sanders Peirce as having created a theory of information in his works on semiotics.^{[42]}^{: 171 }^{[43]}^{: 137 } Nauta defined semiotic information theory as the study of "*the internal processes of coding, filtering, and information processing.*"^{[42]}^{: 91 }

Concepts from information theory such as redundancy and code control have been used by semioticians such as Umberto Eco and Ferruccio Rossi-Landi to explain ideology as a form of message transmission whereby a dominant social class emits its message by using signs that exhibit a high degree of redundancy such that only one message is decoded among a selection of competing ones.^{[44]}

### Integrated process organization of neural information

Quantitative information theoretic methods have been applied in

^{[50]}

^{[51]}

^{[52]}

^{[53]}

^{[54]}

### Miscellaneous applications

Information theory also has applications in the search for extraterrestrial intelligence,^{[55]} black holes, bioinformatics^{[citation needed]}, and gambling.

## See also

- Algorithmic probability
- Bayesian inference
- Communication theory
- Constructor theory – a generalization of information theory that includes quantum information
- Formal science
- Inductive probability
- Info-metrics
- Minimum message length
- Minimum description length
- Philosophy of information

### Applications

- Active networking
- Cryptanalysis
- Cryptography
- Cybernetics
- Entropy in thermodynamics and information theory
- Gambling
- Intelligence (information gathering)
- Seismic exploration

### History

- Hartley, R.V.L.
- History of information theory
- Shannon, C.E.
- Timeline of information theory
- Yockey, H.P.
- Andrey Kolmogorov

### Theory

- Coding theory
- Detection theory
- Estimation theory
- Fisher information
- Information algebra
- Information asymmetry
- Information field theory
- Information geometry
- Information theory and measure theory
- Kolmogorov complexity
- List of unsolved problems in information theory
- Logic of information
- Network coding
- Philosophy of information
- Quantum information science
- Source coding

### Concepts

- Ban (unit)
- Channel capacity
- Communication channel
- Communication source
- Conditional entropy
- Covert channel
- Data compression
- Decoder
- Differential entropy
- Fungible information
- Information fluctuation complexity
- Information entropy
- Joint entropy
- Kullback–Leibler divergence
- Mutual information
- Pointwise mutual information (PMI)
- Receiver (information theory)
- Redundancy
- Rényi entropy
- Self-information
- Unicity distance
- Variety
- Hamming distance
- Perplexity

## References

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- ^ .
- ^ .
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**.****.**- ^ .
**.****.****.****.****.****on 2007-10-07. Retrieved 2008-03-11.****^**David R. Anderson (November 1, 2003). "Some background on why people in the empirical sciences may want to better understand the information-theoretic methods" (PDF). Archived from the original (PDF) on July 23, 2011. Retrieved 2010-06-23.- ISBN 978-3-319-47337-6, retrieved 2024-09-19
**.****.****].****.****.****.****.**- IEEE. Retrieved 2023-09-30.
**. Retrieved 2023-09-30.**- ^
^{a}^{b}Tse, David (2020-12-22). "How Claude Shannon Invented the Future".*Quanta Magazine*. Retrieved 2023-09-30. **^**Braverman, Mark (September 19, 2011). "Information Theory in Computer Science" (PDF).- ISBN 0-486-68210-2.
**.**- ^
**.****.****.****.****.****from the original on Apr 12, 2024 – via TU Delft Repositories.****.****.****.****.****^**Peres, A.; P. F. Scudo (2002b). A. Khrennikov (ed.).*Quantum Theory: Reconsideration of Foundations*. Växjö University Press, Växjö, Sweden. p. 283.**.**- ^ .
**^**Nöth, Winfried (January 2012). "Charles S. Peirce's theory of information: a theory of the growth of symbols and of knowledge".*Cybernetics and Human Knowing*.**19**(1–2): 137–161.**^**Nöth, Winfried (1981). "Semiotics of ideology".*Semiotica*, Issue 148.**^**Maurer, H. (2021). Cognitive Science: Integrative Synchronization Mechanisms in Cognitive Neuroarchitectures of the Modern Connectionism. CRC Press, Boca Raton/FL, chap. 10, ISBN 978-1-351-04352-6. https://doi.org/10.1201/9781351043526**^**Edelman, G.M. and G. Tononi (2000). A Universe of Consciousness: How Matter Becomes Imagination. Basic Books, New York.**^**Tononi, G. and O. Sporns (2003). Measuring information integration. BMC Neuroscience 4: 1-20.**^**Tononi, G. (2004a). An information integration theory of consciousness. BMC Neuroscience 5: 1-22.**^**Tononi, G. (2004b). Consciousness and the brain: theoretical aspects. In: G. Adelman and B. Smith [eds.]: Encyclopedia of Neuroscience. 3rd Ed. Elsevier, Amsterdam, Oxford.**^**Friston, K. and K.E. Stephan (2007). Free-energy and the brain. Synthese 159: 417-458.**^**Friston, K. (2010). The free-energy principle: a unified brain theory. Nature Reviews Neuroscience 11: 127-138.**^**Friston, K., M. Breakstear and G. Deco (2012). Perception and self-organized instability. Frontiers in Computational Neuroscience 6: 1-19.**^**Friston, K. (2013). Life as we know it. Journal of the Royal Society Interface 10: 20130475.**^**Kirchhoff, M., T. Parr, E. Palacios, K. Friston and J. Kiverstein. (2018). The Markov blankets of life: autonomy, active inference and the free energy principle. Journal of the Royal Society Interface 15: 20170792.**.**

**
**## Further reading

### The classic work

- 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. - R.V.L. Hartley, "Transmission of Information",
*Bell System Technical Journal*, July 1928 - Andrey Kolmogorov (1968), "Three approaches to the quantitative definition of information" in
*International Journal of Computer Mathematics*, 2, pp. 157–168.

### Other journal articles

- J. L. Kelly Jr., Princeton, "A New Interpretation of Information Rate"
*Bell System Technical Journal*, Vol. 35, July 1956, pp. 917–26. - R. Landauer, IEEE.org, "Information is Physical"
*Proc. Workshop on Physics and Computation PhysComp'92*(IEEE Comp. Sci.Press, Los Alamitos, 1993) pp. 1–4. - Landauer, R. (1961). "Irreversibility and Heat Generation in the Computing Process" (PDF).
*IBM J. Res. Dev*.**5**(3): 183–191. . - Timme, Nicholas; Alford, Wesley; Flecker, Benjamin; Beggs, John M. (2012). "Multivariate information measures: an experimentalist's perspective". ].

### Textbooks on information theory

- Alajaji, F. and Chen, P.N. An Introduction to Single-User Information Theory. Singapore: Springer, 2018.
- Arndt, C.
*Information Measures, Information and its Description in Science and Engineering*(Springer Series: Signals and Communication Technology), 2004, - Ash, RB.
*Information Theory*. New York: Interscience, 1965. - Goldman, S.
*Information Theory*. New York: Prentice Hall, 1953. New York: Dover 1968 - .
- Mansuripur, M.
*Introduction to Information Theory*. New York: Prentice Hall, 1987. - Pierce, JR. "An introduction to information theory: symbols, signals and noise". Dover (2nd Edition). 1961 (reprinted by Dover 1980).
- ISBN 0-486-68210-2
- .
- Stone, JV. Chapter 1 of book "Information Theory: A Tutorial Introduction", University of Sheffield, England, 2014. ISBN 978-0956372857.
- Yeung, RW.
*A First Course in Information Theory*Kluwer Academic/Plenum Publishers, 2002.ISBN 0-306-46791-7. - Yeung, RW.
*Information Theory and Network Coding*Springer 2008, 2002.ISBN 978-0-387-79233-0

### Other books

- Leon Brillouin,
*Science and Information Theory*, Mineola, N.Y.: Dover, [1956, 1962] 2004. - A. I. Khinchin,
*Mathematical Foundations of Information Theory*, New York: Dover, 1957. - H. S. Leff and A. F. Rex, Editors,
*Maxwell's Demon: Entropy, Information, Computing*, Princeton University Press, Princeton, New Jersey (1990). - Robert K. Logan.
*What is Information? - Propagating Organization in the Biosphere, the Symbolosphere, the Technosphere and the Econosphere*, Toronto: DEMO Publishing. - Tom Siegfried,
*The Bit and the Pendulum*, Wiley, 2000.ISBN 0-471-32174-5 - Jeremy Campbell,
- Henri Theil,
*Economics and Information Theory*, Rand McNally & Company - Chicago, 1967. - Escolano, Suau, Bonev,
*Information Theory in Computer Vision and Pattern Recognition*, Springer, 2009.ISBN 978-1-84882-296-2 - Vlatko Vedral,
*Decoding Reality: The Universe as Quantum Information*, Oxford University Press 2010.

*
*## External links

- "Information",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Lambert F. L. (1999), "Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? Nonsense!",
*Journal of Chemical Education* - IEEE Information Theory Society and ITSOC Monographs, Surveys, and Reviews Archived 2018-06-12 at the Wayback Machine