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{{short description|Scientific study of digital information}}
{{distinguish|Information science}}

{{Information theory}}

'''Information theory''' is the mathematical study of the [[quantification (science)|quantification]], [[Data storage|storage]], and [[telecommunications|communication]] of [[information]]. The field was established and formalized by [[Claude Shannon]] in the 1940s,<ref>{{Cite journal |last=Schneider |first=Thomas D. |date=2006 |title=Claude Shannon: Biologist |journal=IEEE Engineering in Medicine and Biology Magazine: The Quarterly Magazine of the Engineering in Medicine & Biology Society |volume=25 |issue=1 |pages=30–33 |doi=10.1109/memb.2006.1578661 |issn=0739-5175 |pmc=1538977 |pmid=16485389}}</ref> 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]], [[Neuroscience|neurobiology]], [[physics]], and [[electrical engineering]].<ref name=":2">{{Cite journal |last1=Cruces |first1=Sergio |last2=Martín-Clemente |first2=Rubén |last3=Samek |first3=Wojciech |date=2019-07-03 |title=Information Theory Applications in Signal Processing |journal=Entropy |volume=21 |issue=7 |pages=653 |doi=10.3390/e21070653 |doi-access=free |issn=1099-4300 |pmc=7515149 |pmid=33267367|bibcode=2019Entrp..21..653C }}</ref><ref name=":0">{{Cite book |url=https://books.google.com/books?id=TNpVEAAAQBAJ&dq=how+important+is+information+theory+internet+black+holes&pg=PA23 |title=Fractional Order Systems and Applications in Engineering |publisher=Academic Press |year=2023 |isbn=978-0-323-90953-2 |editor-last=Baleanu |editor-first=D. |series=Advanced Studies in Complex Systems |location=London, United Kingdom |pages=23 |language=en |oclc=on1314337815 |editor-last2=Balas |editor-first2=Valentina Emilia |editor-last3=Agarwal |editor-first3=Praveen}}</ref>

A key measure in information theory is [[information entropy|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|fair]] [[coin flip]] (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a [[dice|die]] (which has six equally likely outcomes). Some other important measures in information theory are [[mutual information]], [[channel capacity]], [[error exponent]]s, 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 (file format)|ZIP files]]), and channel coding/[[error detection and correction]] (e.g. for [[digital subscriber line|DSL]]). Its impact has been crucial to the success of the [[Voyager program|Voyager]] missions to deep space,<ref name=":22">{{Cite web |last=Horgan |first=John |author-link=John Horgan (journalist) |date=2016-04-27 |title=Claude Shannon: Tinkerer, Prankster, and Father of Information Theory |url=https://spectrum.ieee.org/claude-shannon-tinkerer-prankster-and-father-of-information-theory |access-date=2024-11-08 |website=[[IEEE]] |language=en}}</ref> the invention of the [[compact disc]], the feasibility of mobile phones and the development of the [[Internet]] and [[artificial intelligence]].<ref>{{Cite book |last=Shi |first=Zhongzhi |url=https://books.google.com/books?id=xMTFCgAAQBAJ |title=Advanced Artificial Intelligence |date=2011 |publisher=[[World Scientific Publishing]] |isbn=978-981-4291-34-7 |pages=2 |language=en |doi=10.1142/7547}}</ref><ref>{{Cite book |last1=Sinha |first1=Sudhi |url=https://books.google.com/books?id=2pb-DwAAQBAJ |title=Reimagining Businesses with AI |last2=Al Huraimel |first2=Khaled |date=2020-10-20 |publisher=Wiley |isbn=978-1-119-70915-2 |edition=1 |pages=4 |language=en |doi=10.1002/9781119709183}}</ref><ref name=":0" /> The theory has also found applications in other areas, including [[statistical inference]],<ref>{{cite book|last1=Burnham|first1=K. P.|last2=Anderson|first2=D. R.|year=2002|title=Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach|edition=Second|language=en|publisher=Springer Science|location=New York|isbn=978-0-387-95364-9}}</ref> [[cryptography]], [[neurobiology]],<ref name="Spikes">{{cite book|title=Spikes: Exploring the Neural Code|author1=F. Rieke|author2=D. Warland|author3=R Ruyter van Steveninck|author4=W Bialek|publisher=The MIT press|year=1997|isbn=978-0262681087}}</ref> [[perception]],<ref>{{Cite journal|last1=Delgado-Bonal|first1=Alfonso|last2=Martín-Torres|first2=Javier|date=2016-11-03|title=Human vision is determined based on information theory|journal=Scientific Reports|language=En|volume=6|issue=1|pages=36038|bibcode=2016NatSR...636038D|doi=10.1038/srep36038|issn=2045-2322|pmc=5093619|pmid=27808236}}</ref> [[signal processing]],<ref name=":2" /> [[linguistics]], the evolution<ref>{{cite journal|last1=cf|last2=Huelsenbeck|first2=J. P.|last3=Ronquist|first3=F.|last4=Nielsen|first4=R.|last5=Bollback|first5=J. P.|year=2001|title=Bayesian inference of phylogeny and its impact on evolutionary biology|journal=Science|volume=294|issue=5550|pages=2310–2314|bibcode=2001Sci...294.2310H|doi=10.1126/science.1065889|pmid=11743192|s2cid=2138288}}</ref> and function<ref>{{cite journal|last1=Allikmets|first1=Rando|last2=Wasserman|first2=Wyeth W.|last3=Hutchinson|first3=Amy|last4=Smallwood|first4=Philip|last5=Nathans|first5=Jeremy|last6=Rogan|first6=Peter K.|year=1998|title=Thomas D. Schneider], Michael Dean (1998) Organization of the ABCR gene: analysis of promoter and splice junction sequences|url=http://alum.mit.edu/www/toms/|journal=Gene|volume=215|issue=1|pages=111–122|doi=10.1016/s0378-1119(98)00269-8|pmid=9666097|doi-access=free}}</ref> of molecular codes ([[bioinformatics]]), [[thermal physics]],<ref>{{cite journal|last1=Jaynes|first1=E. T.|year=1957|title=Information Theory and Statistical Mechanics|url=http://bayes.wustl.edu/|journal=Phys. Rev.|volume=106|issue=4|page=620|bibcode=1957PhRv..106..620J|doi=10.1103/physrev.106.620|s2cid=17870175 }}</ref> [[molecular dynamics]],<ref>{{Cite journal|last1=Talaat|first1=Khaled|last2=Cowen|first2=Benjamin|last3=Anderoglu|first3=Osman|date=2020-10-05|title=Method of information entropy for convergence assessment of molecular dynamics simulations|journal=Journal of Applied Physics|language=En|volume=128|issue=13|pages=135102|doi=10.1063/5.0019078|bibcode=2020JAP...128m5102T|osti=1691442|s2cid=225010720|doi-access=free}}</ref> [[black hole]]s, [[quantum computing]], [[information retrieval]], [[Intelligence (Information Gathering)|intelligence gathering]], [[plagiarism detection]],<ref>{{cite journal|last1=Bennett|first1=Charles H.|last2=Li|first2=Ming|last3=Ma|first3=Bin|year=2003|title=Chain Letters and Evolutionary Histories|url=http://sciamdigital.com/index.cfm?fa=Products.ViewIssuePreview&ARTICLEID_CHAR=08B64096-0772-4904-9D48227D5C9FAC75|journal=[[Scientific American]]|volume=288|issue=6|pages=76–81|bibcode=2003SciAm.288f..76B|doi=10.1038/scientificamerican0603-76|pmid=12764940|access-date=2008-03-11|archive-url=https://web.archive.org/web/20071007041539/http://www.sciamdigital.com/index.cfm?fa=Products.ViewIssuePreview&ARTICLEID_CHAR=08B64096-0772-4904-9D48227D5C9FAC75|archive-date=2007-10-07|url-status=dead}}</ref> [[pattern recognition]], [[anomaly detection]],<ref>{{Cite web|url=http://aicanderson2.home.comcast.net/~aicanderson2/home.pdf|title=Some background on why people in the empirical sciences may want to better understand the information-theoretic methods|author=David R. Anderson|date=November 1, 2003|archive-url=https://web.archive.org/web/20110723045720/http://aicanderson2.home.comcast.net/~aicanderson2/home.pdf|archive-date=July 23, 2011|url-status=dead|access-date=2010-06-23}}
</ref> the analysis of [[music]],<ref>{{Citation |last=Loy |first=D. Gareth |title=Music, Expectation, and Information Theory |date=2017 |work=The Musical-Mathematical Mind: Patterns and Transformations |series=Computational Music Science |pages=161–169 |editor-last=Pareyon |editor-first=Gabriel |url=https://link.springer.com/chapter/10.1007/978-3-319-47337-6_17 |access-date=2024-09-19 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-47337-6_17 |isbn=978-3-319-47337-6 |editor2-last=Pina-Romero |editor2-first=Silvia |editor3-last=Agustín-Aquino |editor3-first=Octavio A. |editor4-last=Lluis-Puebla |editor4-first=Emilio}}</ref><ref>{{Cite journal |last1=Rocamora |first1=Martín |last2=Cancela |first2=Pablo |last3=Biscainho |first3=Luiz |date=2019-04-05 |title=Information Theory Concepts Applied to the Analysis of Rhythm in Recorded Music with Recurrent Rhythmic Patterns |url=http://www.aes.org/e-lib/browse.cfm?elib=20449 |journal=Journal of the Audio Engineering Society |volume=67 |issue=4 |pages=160–173 |doi=10.17743/jaes.2019.0003}}</ref> [[Art|art creation]],<ref>{{Cite journal |last=Marsden |first=Alan |date=2020 |title=New Prospects for Information Theory in Arts Research |url=https://direct.mit.edu/leon/article/53/3/274-280/96875 |journal=Leonardo |language=en |volume=53 |issue=3 |pages=274–280 |doi=10.1162/leon_a_01860 |issn=0024-094X}}</ref> [[imaging system]] design,<ref>{{Cite arXiv|title=Universal evaluation and design of imaging systems using information estimation|eprint=2405.20559 |last1=Pinkard |first1=Henry |last2=Kabuli |first2=Leyla |last3=Markley |first3=Eric |last4=Chien |first4=Tiffany |last5=Jiao |first5=Jiantao |last6=Waller |first6=Laura |date=2024 |class=physics.optics }}</ref> study of [[outer space]],<ref>{{Cite journal |last1=Wing |first1=Simon |last2=Johnson |first2=Jay R. |date=2019-02-01 |title=Applications of Information Theory in Solar and Space Physics |journal=Entropy |language=en |volume=21 |issue=2 |pages=140 |doi=10.3390/e21020140 |issn=1099-4300 |pmc=7514618 |pmid=33266856 |doi-access=free|bibcode=2019Entrp..21..140W }}</ref> the dimensionality of [[space]],<ref>{{Cite journal |last=Kak |first=Subhash |date=2020-11-26 |title=Information theory and dimensionality of space |journal=Scientific Reports |language=en |volume=10 |issue=1 |pages=20733 |doi=10.1038/s41598-020-77855-9 |pmid=33244156 |pmc=7693271 |issn=2045-2322}}</ref> and [[epistemology]].<ref>{{Cite journal |last=Harms |first=William F. |date=1998 |title=The Use of Information Theory in Epistemology |url=https://www.jstor.org/stable/188281 |journal=Philosophy of Science |volume=65 |issue=3 |pages=472–501 |doi=10.1086/392657 |jstor=188281 |issn=0031-8248}}</ref>

==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.<ref name="Spikes" />

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.{{citation needed|date=April 2024}}

A third class of information theory codes are [[cryptographic algorithm]]s (both [[code (cryptography)|code]]s and [[cipher]]s). Concepts, methods and results from coding theory and information theory are widely used in cryptography and [[cryptanalysis]], such as the [[Ban (unit)|unit ban]].{{citation needed|date=April 2024}}

==Historical background==
{{Main|History of information theory}}

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, noting that the paper was "even more profound and more fundamental" than the [[transistor]].{{sfn|Gleick|2011|pp=3–4}} He came to be known as the "father of information theory".<ref>{{Cite web |last=Horgan |first=John |date=2016-04-27 |title=Claude Shannon: Tinkerer, Prankster, and Father of Information Theory |url=https://spectrum.ieee.org/claude-shannon-tinkerer-prankster-and-father-of-information-theory |access-date=2023-09-30 |website=[[IEEE]] |language=en}}</ref><ref>{{Cite magazine |last=Roberts |first=Siobhan |date=2016-04-30 |title=The Forgotten Father of the Information Age |language=en-US |magazine=The New Yorker |url=https://www.newyorker.com/tech/annals-of-technology/claude-shannon-the-father-of-the-information-age-turns-1100100 |access-date=2023-09-30 |issn=0028-792X}}</ref><ref name=":1">{{Cite web |last=Tse |first=David |date=2020-12-22 |title=How Claude Shannon Invented the Future |url=https://www.quantamagazine.org/how-claude-shannons-information-theory-invented-the-future-20201222/ |access-date=2023-09-30 |website=Quanta Magazine}}</ref> Shannon outlined some of his initial ideas of information theory as early as 1939 in a letter to [[Vannevar Bush]].<ref name=":1" />

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 {{math|1=''W'' = ''K'' log ''m''}} (recalling the [[Boltzmann 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 {{math|1=''H'' = log ''S''<sup>''n''</sup> = ''n'' log ''S''}}, 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 (unit)|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 [[Cryptanalysis of the Enigma|Enigma]] ciphers.{{citation needed|date=April 2024}}

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]]''.{{citation needed|date=April 2024}}

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 (information theory)|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.{{citation needed|date=April 2024}}

==Quantities of information==
{{Unreferenced section|date=April 2024}}{{Main|Quantities of information}}

Information theory is based on [[probability theory]] and statistics, where [[Quantities of information|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 (information theory)|entropy]], which forms the building block of many other measures. Entropy allows quantification of measure of information in a single random variable.<ref>{{Cite web |last=Braverman |first=Mark |date=September 19, 2011 |title=Information Theory in Computer Science |url=https://www.cs.princeton.edu/courses/archive/fall11/cos597D/L01.pdf }}</ref> 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 [[Communication channel|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 [[units of measurement|unit]] of information entropy that is used. A common unit of information is the bit or [[shannon (unit)|shannon]], based on the [[binary logarithm]]. Other units include the [[nat (unit)|nat]], which is based on the [[natural logarithm]], and the [[deciban|decimal digit]], which is based on the [[common logarithm]].

In what follows, an expression of the form {{math|''p'' log ''p''}} is considered by convention to be equal to zero whenever {{math|1=''p'' = 0}}. This is justified because <math>\lim_{p \rightarrow 0+} p \log p = 0</math> 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 (information theory)|entropy]] {{math|''H''}}, in units of bits (per symbol), is given by
:<math>H = - \sum_{i} p_i \log_2 (p_i)</math>
where {{math|''p<sub>i</sub>''}} is the probability of occurrence of the {{math|''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 (unit)|shannon]] in his honor. Entropy is also commonly computed using the natural logarithm (base {{mvar|[[E (mathematical constant)|e]]}}, where {{mvar|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 {{nowrap|1=2<sup>8</sup> = 256}} will produce a measurement in [[byte]]s per symbol, and a logarithm of base 10 will produce a measurement in decimal digits (or [[Hartley (unit)|hartleys]]) per symbol.

Intuitively, the entropy {{math|''H<sub>X</sub>''}} of a discrete random variable {{math|''X''}} is a measure of the amount of ''uncertainty'' associated with the value of {{math|''X''}} when only its distribution is known.

The entropy of a source that emits a sequence of {{math|''N''}} symbols that are [[independent and identically distributed]] (iid) is {{math|''N'' ⋅ ''H''}} bits (per message of {{math|''N''}} symbols). If the source data symbols are identically distributed but not independent, the entropy of a message of length {{math|''N''}} will be less than {{math|''N'' ⋅ ''H''}}.

[[File:Binary entropy plot.svg|thumbnail|right|200px|The entropy of a [[Bernoulli trial]] as a function of success probability, often called the {{em|[[binary entropy function]]}}, {{math|''H''<sub>b</sub>(''p'')}}. The entropy is maximized at 1 bit per trial when the two possible outcomes are equally probable, as in an unbiased coin toss.]]

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 <math>\mathbb{X}</math> is the set of all messages {{math|{{mset|''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}}}} that {{math|''X''}} could be, and {{math|''p''(''x'')}} is the probability of some <math>x \in \mathbb X</math>, then the entropy, {{math|''H''}}, of {{math|''X''}} is defined:{{sfn|Reza|1994}}

:<math> H(X) = \mathbb{E}_{X} [I(x)] = -\sum_{x \in \mathbb{X}} p(x) \log p(x).</math>

(Here, {{math|''I''(''x'')}} is the [[self-information]], which is the entropy contribution of an individual message, and <math>\mathbb{E}_X</math> is the [[expected value]].) A property of entropy is that it is maximized when all the messages in the message space are equiprobable {{math|1=''p''(''x'') = 1/''n''}}; i.e., most unpredictable, in which case {{math|1=''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 (unit)|shannon]] (Sh) as unit:

:<math>H_{\mathrm{b}}(p) = - p \log_2 p - (1-p)\log_2 (1-p).</math>

===Joint entropy===
The {{em|[[joint entropy]]}} of two discrete random variables {{math|''X''}} and {{math|''Y''}} is merely the entropy of their pairing: {{math|(''X'', ''Y'')}}. This implies that if {{math|''X''}} and {{math|''Y''}} are [[statistical independence|independent]], then their joint entropy is the sum of their individual entropies.

For example, if {{math|(''X'', ''Y'')}} represents the position of a chess piece—{{math|''X''}} the row and {{math|''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.

:<math>H(X, Y) = \mathbb{E}_{X,Y} [-\log p(x,y)] = - \sum_{x, y} p(x, y) \log p(x, y) \,</math>

Despite similar notation, joint entropy should not be confused with {{em|[[cross-entropy]]}}.

===Conditional entropy (equivocation)===
The {{em|[[conditional entropy]]}} or ''conditional uncertainty'' of {{math|''X''}} given random variable {{math|''Y''}} (also called the ''equivocation'' of {{math|''X''}} about {{math|''Y''}}) is the average conditional entropy over {{math|''Y''}}:{{sfn|Ash|1990}}

:<math> 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 p(x|y).</math>

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:

: <math> H(X|Y) = H(X,Y) - H(Y) .\,</math>

===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 {{math|''X''}} relative to {{math|''Y''}} is given by:

:<math>I(X;Y) = \mathbb{E}_{X,Y} [SI(x,y)] = \sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)\, p(y)}</math>
where {{math|SI}} (''S''pecific mutual Information) is the [[pointwise mutual information]].

A basic property of the mutual information is that
: <math>I(X;Y) = H(X) - H(X|Y).\,</math>
That is, knowing ''Y'', we can save an average of {{math|''I''(''X''; ''Y'')}} bits in encoding ''X'' compared to not knowing ''Y''.

Mutual information is [[symmetric function|symmetric]]:
: <math>I(X;Y) = I(Y;X) = H(X) + H(Y) - H(X,Y).\,</math>

Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) between the [[posterior probability|posterior probability distribution]] of ''X'' given the value of ''Y'' and the [[prior probability|prior distribution]] on ''X'':
: <math>I(X;Y) = \mathbb E_{p(y)} [D_{\mathrm{KL}}( p(X|Y=y) \| p(X) )].</math>
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:
: <math>I(X; Y) = D_{\mathrm{KL}}(p(X,Y) \| p(X)p(Y)).</math>

Mutual information is closely related to the [[likelihood-ratio test|log-likelihood ratio test]] in the context of contingency tables and the [[multinomial distribution]] and to [[Pearson's chi-squared test|Pearson's χ<sup>2</sup> 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]] {{tmath|p(X)}}, and an arbitrary probability distribution {{tmath|q(X)}}. If we compress data in a manner that assumes {{tmath|q(X)}} is the distribution underlying some data, when, in reality, {{tmath|p(X)}} is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus defined

:<math>D_{\mathrm{KL}}(p(X) \| q(X)) = \sum_{x \in X} -p(x) \log {q(x)} \, - \, \sum_{x \in X} -p(x) \log {p(x)} = \sum_{x \in X} p(x) \log \frac{p(x)}{q(x)}.</math>

Although it is sometimes used as a 'distance metric', KL divergence is not a true [[Metric (mathematics)|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 {{tmath|p(x)}}. If Alice knows the true distribution {{tmath|p(x)}}, while Bob believes (has a [[prior probability|prior]]) that the distribution is {{tmath|q(x)}}, then Bob will be more [[Information content|surprised]] than Alice, on average, upon seeing the value of ''X''. The KL divergence is the (objective) expected value of Bob's (subjective) [[Information content|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]]'', <math>I(X^n\to Y^n) </math>, is an information theory measure that quantifies the [[information]] flow from the random process <math>X^n = \{X_1,X_2,\dots,X_n\}</math> to the random process <math>Y^n = \{Y_1,Y_2,\dots,Y_n\}</math>. The term ''directed information'' was coined by [[James Massey]] and is defined as
:<math>I(X^n\to Y^n) \triangleq \sum_{i=1}^n I(X^i;Y_i|Y^{i-1})</math>,
where <math>I(X^{i};Y_i|Y^{i-1})</math> is the [[conditional mutual information]] <math>I(X_1,X_2,...,X_{i};Y_i|Y_1,Y_2,...,Y_{i-1})</math>.

In contrast to ''mutual'' information, ''directed'' information is not symmetric. The <math>I(X^n\to Y^n) </math> measures the information bits that are transmitted causally{{clarify|date=November 2024|reason=definition of causal transmission?}} from <math>X^n</math> to <math>Y^n</math>. The Directed information has many applications in problems where [[causality]] plays an important role such as [[channel capacity|capacity of channel]] with feedback,<ref name=massey>{{citation |last1=Massey|first1=James|contribution=Causality, Feedback And Directed Information|date=1990|title=Proc. 1990 Intl. Symp. on Info. Th. and its Applications|citeseerx=10.1.1.36.5688}}</ref><ref>{{cite journal|last1=Permuter|first1=Haim Henry|last2=Weissman|first2=Tsachy|last3=Goldsmith|first3=Andrea J.|title=Finite State Channels With Time-Invariant Deterministic Feedback|journal=IEEE Transactions on Information Theory|date=February 2009|volume=55|issue=2|pages=644–662|doi=10.1109/TIT.2008.2009849|arxiv=cs/0608070|s2cid=13178}}</ref> capacity of discrete [[memoryless]] networks with feedback,<ref>{{cite journal|last1=Kramer|first1=G.|title=Capacity results for the discrete memoryless network|journal=IEEE Transactions on Information Theory|date=January 2003|volume=49|issue=1|pages=4–21|doi=10.1109/TIT.2002.806135}}</ref> [[sports gambling|gambling]] with causal side information,<ref>{{cite journal|last1=Permuter|first1=Haim H.|last2=Kim|first2=Young-Han|last3=Weissman|first3=Tsachy|title=Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing|journal=IEEE Transactions on Information Theory|date=June 2011|volume=57|issue=6|pages=3248–3259|doi=10.1109/TIT.2011.2136270|arxiv=0912.4872|s2cid=11722596}}</ref> [[Data compression|compression]] with causal side information,<ref>{{cite journal|last1=Simeone|first1=Osvaldo|last2=Permuter|first2=Haim Henri|title=Source Coding When the Side Information May Be Delayed|journal=IEEE Transactions on Information Theory|date=June 2013|volume=59|issue=6|pages=3607–3618|doi=10.1109/TIT.2013.2248192|arxiv=1109.1293|s2cid=3211485}}</ref>
[[real-time control]] communication settings,<ref>{{cite journal|last1=Charalambous|first1=Charalambos D.|last2=Stavrou|first2=Photios A.|title=Directed Information on Abstract Spaces: Properties and Variational Equalities|journal=IEEE Transactions on Information Theory|date=August 2016|volume=62|issue=11|pages=6019–6052|doi=10.1109/TIT.2016.2604846|arxiv=1302.3971|s2cid=8107565}}</ref><ref>{{cite journal |last1=Tanaka |first1=Takashi |last2=Esfahani |first2=Peyman Mohajerin |last3=Mitter |first3=Sanjoy K. |title=LQG Control With Minimum Directed Information: Semidefinite Programming Approach |journal=IEEE Transactions on Automatic Control |date=January 2018 |volume=63 |issue=1 |pages=37–52 |doi=10.1109/TAC.2017.2709618|s2cid=1401958 |url=http://resolver.tudelft.nl/uuid:d9db1c11-fbfd-4c0c-b66f-f341b49fa61a |arxiv=1510.04214 |via=TU Delft Repositories |url-status=live |archive-url=https://web.archive.org/web/20240412014101/https://repository.tudelft.nl/islandora/object/uuid:d9db1c11-fbfd-4c0c-b66f-f341b49fa61a/datastream/OBJ/download |archive-date= Apr 12, 2024  }}</ref> and in statistical physics.<ref>{{cite journal |last1=Vinkler |first1=Dror A |last2=Permuter |first2=Haim H |last3=Merhav |first3=Neri |title=Analogy between gambling and measurement-based work extraction |journal=Journal of Statistical Mechanics: Theory and Experiment |date=20 April 2016 |volume=2016 |issue=4 |pages=043403 |doi=10.1088/1742-5468/2016/04/043403|arxiv=1404.6788 |bibcode=2016JSMTE..04.3403V |s2cid=124719237 }}</ref>

===Other quantities===
Other important information theoretic quantities include the [[Rényi entropy]] and the [[Tsallis entropy]] (generalizations of the concept of entropy), [[differential entropy]] (a generalization of quantities of information to continuous distributions), and the [[conditional mutual information]]. Also, [[pragmatic information]] has been proposed as a measure of how much information has been used in making a decision.

==Coding theory==
{{Unreferenced section|date=April 2024}}{{Main|Coding theory}}

[[File:CDSCRATCHES.jpg|thumb|right|A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using [[error detection and correction]].]]

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 code]]s (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 [[broadcast channel]]) or intermediary "helpers" (the [[relay channel]]), or more general [[computer network|networks]], compression followed by transmission may no longer be optimal.

===Source theory===
Any process that generates successive messages can be considered a {{em|[[Communication source|source]]}} of information. A memoryless source is one in which each message is an [[Independent identically distributed random variables|independent identically distributed random variable]], whereas the properties of [[ergodic theory|ergodicity]] and [[stationary process|stationarity]] impose less restrictive constraints. All such sources are [[stochastic process|stochastic]]. These terms are well studied in their own right outside information theory.

====Rate====<!-- This section is linked from [[Channel capacity]] -->
Information ''[[Entropy rate|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:

:<math>r = \lim_{n \to \infty} H(X_n|X_{n-1},X_{n-2},X_{n-3}, \ldots);</math>

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:

:<math>r = \lim_{n \to \infty} \frac{1}{n} H(X_1, X_2, \dots X_n);</math>

that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result.<ref>{{cite book | title = Digital Compression for Multimedia: Principles and Standards | author = Jerry D. Gibson | publisher = Morgan Kaufmann | year = 1998 | url = https://books.google.com/books?id=aqQ2Ry6spu0C&q=entropy-rate+conditional&pg=PA56 | isbn = 1-55860-369-7 }}</ref>

The [[information rate]] is defined as:  
:<math>r = \lim_{n \to \infty} \frac{1}{n} I(X_1, X_2, \dots X_n;Y_1,Y_2, \dots Y_n);</math>

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 {{em|source coding}}.

===Channel capacity===
{{Main|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:

:<math title="Channel model">
\xrightarrow[\text{Message}]{W}
\begin{array}{ |c| }\hline \text{Encoder} \\ f_n \\ \hline\end{array} \xrightarrow[\mathrm{Encoded \atop sequence}]{X^n} \begin{array}{ |c| }\hline \text{Channel} \\ p(y|x) \\ \hline\end{array} \xrightarrow[\mathrm{Received \atop sequence}]{Y^n} \begin{array}{ |c| }\hline \text{Decoder} \\ g_n \\ \hline\end{array} \xrightarrow[\mathrm{Estimated \atop message}]{\hat W}</math>

Here ''X'' represents the space of messages transmitted, and ''Y'' the space of messages received during a unit time over our channel. Let {{math|''p''(''y''{{pipe}}''x'')}} be the [[conditional probability]] distribution function of ''Y'' given ''X''. We will consider {{math|''p''(''y''{{pipe}}''x'')}} to be an inherent fixed property of our communications channel (representing the nature of the ''[[Signal noise|noise]]'' of our channel). Then the joint distribution of ''X'' and ''Y'' is completely determined by our channel and by our choice of {{math|''f''(''x'')}}, the marginal distribution of messages we choose to send over the channel. Under these constraints, we would like to maximize the rate of information, or the ''[[Signal (electrical engineering)|signal]]'', we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the {{em|channel capacity}} and is given by:
:<math> C = \max_{f} I(X;Y).\! </math>
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'' &gt; ''C'', it is impossible to transmit with arbitrarily small block error.

''[[Channel code|Channel coding]]'' 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.

====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 {{math|1 &minus; ''H''<sub>b</sub>(''p'')}} bits per channel use, where {{math|''H''<sub>b</sub>}} is the binary entropy function to the base-2 logarithm:

::[[File:Binary symmetric channel.svg]]

* 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 {{nowrap|1 &minus; ''p''}} bits per channel use.

::[[File:Binary erasure channel.svg]]

====Channels with memory and directed information====
In practice many channels have memory. Namely, at time <math> i </math> the channel is given by the conditional probability<math> P(y_i|x_i,x_{i-1},x_{i-2},...,x_1,y_{i-1},y_{i-2},...,y_1) </math>.
It is often more comfortable to use the notation <math> x^i=(x_i,x_{i-1},x_{i-2},...,x_1) </math> and the channel become <math> P(y_i|x^i,y^{i-1}) </math>.
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<ref name=massey/><ref>{{cite journal |last1=Permuter |first1=Haim Henry |last2=Weissman |first2=Tsachy |last3=Goldsmith |first3=Andrea J. |title=Finite State Channels With Time-Invariant Deterministic Feedback |journal=IEEE Transactions on Information Theory |date=February 2009 |volume=55 |issue=2 |pages=644–662 |doi=10.1109/TIT.2008.2009849|arxiv=cs/0608070 |s2cid=13178 }}</ref> (if there is no feedback the directed information equals the mutual information).

===Fungible information===
'''Fungible information''' is the [[information]] for which the means of [[encoding]] is not important.<ref>{{cite journal|last=Bartlett|first=Stephen D. |author2=Rudolph, Terry|author3-link=Robert Spekkens|author3=Spekkens, Robert W.|title=Reference frames, superselection rules, and quantum information|journal=Reviews of Modern Physics|volume=79|issue=2|date=April–June 2007|pages=555–606|doi=10.1103/RevModPhys.79.555|bibcode=2007RvMP...79..555B|arxiv = quant-ph/0610030 }}</ref> Classical information theorists and computer scientists are mainly concerned with information of this sort. It is sometimes referred as speakable information.<ref>{{cite book | last = Peres | first = A.| title = Quantum Theory: Reconsideration of Foundations | publisher = Växjö University Press, Växjö, Sweden| year = 2002b|editor = A. Khrennikov|page=283 |author2=P. F. Scudo}}</ref>

==Applications to other fields==

===Intelligence uses and secrecy applications===
{{Unreferenced section|date=April 2024}}
Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the [[Ban (unit)|ban]], was used in the [[Ultra (cryptography)|Ultra]] project, breaking the German [[Enigma machine]] code and hastening the [[Victory in Europe Day|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.

Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A [[brute force attack]] can break systems based on [[public-key cryptography|asymmetric key algorithms]] or on most commonly used methods of [[symmetric-key algorithm|symmetric key algorithms]] (sometimes called secret key algorithms), such as [[block cipher]]s. 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 (cryptography)|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.

===Pseudorandom number generation===
{{Unreferenced section|date=April 2024}}
[[Pseudorandom number generator]]s 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 generator]]s, but even they require [[random seed]]s external to the software to work as intended. These can be obtained via [[Extractor (mathematics)|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.<ref>{{cite journal|doi=10.1002/smj.4250020202 | volume=2 | issue=2 | title=The corporation and innovation | year=1981 | journal=Strategic Management Journal | pages=97–118 | last1 = Haggerty | first1 = Patrick E.}}</ref>

===Semiotics===
[[Semiotics|Semioticians]] {{ill|Doede Nauta|nl}} and [[Winfried Nöth]] both considered [[Charles Sanders Peirce]] as having created a theory of information in his works on semiotics.<ref name="Nauta 1972">{{cite book |last1=Nauta |first1=Doede |title=The Meaning of Information |date=1972 |publisher=Mouton |location=The Hague |isbn=9789027919960}}</ref>{{rp|171}}<ref name="Nöth 2012">{{cite journal |last1=Nöth |first1=Winfried |title=Charles S. Peirce's theory of information: a theory of the growth of symbols and of knowledge |journal=Cybernetics and Human Knowing |date=January 2012 |volume=19 |issue=1–2 |pages=137–161 |url=https://edisciplinas.usp.br/mod/resource/view.php?id=2311849}}</ref>{{rp|137}} Nauta defined semiotic information theory as the study of "''the internal processes of coding, filtering, and information processing.''"<ref name="Nauta 1972"/>{{rp|91}}

Concepts from information theory such as redundancy and code control have been used by semioticians such as [[Umberto Eco]] and {{ill|Ferruccio Rossi-Landi|it}} 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.<ref>Nöth, Winfried (1981). "[https://kobra.uni-kassel.de/bitstream/handle/123456789/2014122246977/semi_2004_002.pdf?sequence=1&isAllowed=y Semiotics of ideology]". ''Semiotica'', Issue 148.</ref>

===Integrated process organization of neural information===
Quantitative information theoretic methods have been applied in [[cognitive science]] to analyze the integrated process organization of neural information in the context of the [[binding problem]] in [[cognitive neuroscience]].<ref>{{cite book|last=Maurer|first=H.|year=2021|title=Cognitive Science: Integrative Synchronization Mechanisms in Cognitive Neuroarchitectures of the Modern Connectionism|language=en|publisher=CRC Press|location=Boca Raton/FL|chapter=Chapter 10: Systematic Class of Information Based Architecture Types|isbn=978-1-351-04352-6|doi=10.1201/9781351043526}}</ref> In this context, either an information-theoretical measure, such as {{em|functional clusters}} ([[Gerald Edelman]] and [[Giulio Tononi]]'s functional clustering model and dynamic core hypothesis (DCH)<ref>{{cite book|last1=Edelman|first1=G.M.|first2=G.|last2=Tononi|year=2000|title=A Universe of Consciousness: How Matter Becomes Imagination|language=en|publisher=Basic Books|location=New York|isbn=978-0465013777}}</ref>) or {{em|effective information}} (Tononi's [[integrated information theory]] (IIT) of consciousness<ref>{{cite journal|last1=Tononi|first1=G.|first2=O.|last2=Sporns|year=2003|title=Measuring information integration|journal=BMC Neuroscience|language=en|volume=4|pages=1–20|doi=10.1186/1471-2202-4-31|doi-access=free |pmid=14641936 |pmc=331407 }}</ref><ref>{{cite journal|last=Tononi|first=G.|year=2004a|title=An information integration theory of consciousness|journal=BMC Neuroscience|language=en|volume=5|pages=1–22|doi=10.1186/1471-2202-5-42|doi-access=free |pmid=15522121 |pmc=543470 }}</ref><ref>{{cite book|last=Tononi|first=G.|year=2004b|chapter=Consciousness and the brain: theoretical aspects|editor1-first=G.|editor1-last=Adelman|editor2-first=B.|editor2-last=Smith|title=Encyclopedia of Neuroscience|language=en|edition=3rd|publisher=Elsevier|location=Amsterdam, Oxford|chapter-url=https://www.researchgate.net/publication/265238140|archive-url=https://web.archive.org/web/20231202031406/https://www.jsmf.org/meetings/2003/nov/consciousness_encyclopedia_2003.pdf|archive-date=2023-12-02|url-status=live|isbn=0-444-51432-5}}</ref>), is defined (on the basis of a reentrant process organization, i.e. the synchronization of neurophysiological activity between groups of neuronal populations), or the measure of the minimization of free energy on the basis of statistical methods ([[Karl J. Friston]]'s [[free energy principle]] (FEP), an information-theoretical measure which states that every adaptive change in a self-organized system leads to a minimization of free energy, and the [[Bayesian brain]] hypothesis<ref>{{cite journal|last1=Friston|first1=K.|first2=K.E.|last2=Stephan|year=2007|title=Free-energy and the brain|journal=Synthese|language=en|volume=159|issue=3 |pages=417–458|doi=10.1007/s11229-007-9237-y|pmid=19325932 |pmc=2660582 }}</ref><ref>{{cite journal|last=Friston|first=K.|year=2010|title=The free-energy principle: a unified brain theory|journal=Nature Reviews Neuroscience|language=en|volume=11|issue=2 |pages=127–138|doi=10.1038/nrn2787|pmid=20068583 }}</ref><ref>{{cite journal|last1=Friston|first1=K.|first2=M.|last2=Breakstear|first3=G.|last3=Deco|year=2012|title=Perception and self-organized instability|journal=Frontiers in Computational Neuroscience|language=en|volume=6|pages=1–19|doi=10.3389/fncom.2012.00044|doi-access=free |pmid=22783185 |pmc=3390798 }}</ref><ref>{{cite journal|last=Friston|first=K.|year=2013|title=Life as we know it|journal=Journal of the Royal Society Interface|language=en|volume=10|issue=86 |pages=20130475|doi=10.1098/rsif.2013.0475|pmid=23825119 |pmc=3730701 }}</ref><ref>{{cite journal|last1=Kirchhoff|first1=M.|first2=T.|last2=Parr|first3=E.|last3=Palacios|first4=K.|last4=Friston|first5=J.|last5=Kiverstein|year=2018|title=The Markov blankets of life: autonomy, active inference and the free energy principle|journal=Journal of the Royal Society Interface|language=en|volume=15|issue=138 |pages=20170792|doi=10.1098/rsif.2017.0792|pmid=29343629 |pmc=5805980 }}</ref>).

===Miscellaneous applications===
Information theory also has applications in the [[search for extraterrestrial intelligence]],<ref>{{Cite journal |last1=Doyle |first1=Laurance R. |author-link=Laurance Doyle |last2=McCowan |first2=Brenda |author-link2=Brenda McCowan |last3=Johnston |first3=Simon |last4=Hanser |first4=Sean F. |date=February 2011 |title=Information theory, animal communication, and the search for extraterrestrial intelligence |journal=[[Acta Astronautica]] |language=en |volume=68 |issue=3–4 |pages=406–417 |doi=10.1016/j.actaastro.2009.11.018|bibcode=2011AcAau..68..406D }}</ref> [[black hole information paradox|black holes]],<ref>{{Cite journal |last=Bekenstein |first=Jacob D |date=2004 |title=Black holes and information theory |url=https://www.tandfonline.com/doi/abs/10.1080/00107510310001632523 |journal=Contemporary Physics |volume=45 |issue=1 |pages=31–43 |doi=10.1080/00107510310001632523 |arxiv=quant-ph/0311049 |bibcode=2004ConPh..45...31B |issn=0010-7514}}</ref> [[bioinformatics]],<ref>{{Cite journal |last=Vinga |first=Susana |date=2014-05-01 |title=Information theory applications for biological sequence analysis |url=https://academic.oup.com/bib/article/15/3/376/183705 |journal=Briefings in Bioinformatics |volume=15 |issue=3 |pages=376–389 |doi=10.1093/bib/bbt068 |issn=1467-5463 |pmc=7109941 |pmid=24058049}}</ref> and [[Gambling and information theory|gambling]].<ref>{{Citation |last=Thorp |first=Edward O. |title=The kelly criterion in blackjack sports betting, and the stock market* |date=2008-01-01 |work=Handbook of Asset and Liability Management |pages=385–428 |editor-last=Zenios |editor-first=S. A. |url=https://linkinghub.elsevier.com/retrieve/pii/B9780444532480500150 |access-date=2025-01-20 |place=San Diego |publisher=North-Holland |doi=10.1016/b978-044453248-0.50015-0 |isbn=978-0-444-53248-0 |editor2-last=Ziemba |editor2-first=W. T.}}</ref><ref>{{Cite journal |last=Haigh |first=John |date=2000 |title=The Kelly Criterion and Bet Comparisons in Spread Betting |url=https://rss.onlinelibrary.wiley.com/doi/10.1111/1467-9884.00251 |journal=Journal of the Royal Statistical Society, Series D (The Statistician) |language=en |volume=49 |issue=4 |pages=531–539 |doi=10.1111/1467-9884.00251 |issn=1467-9884}}</ref>

==See also==
{{Portal|Mathematics}}
{{cols|colwidth=21em}}
* [[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]]
{{colend}}

===Applications===
{{cols}}
* [[Active networking]]
* [[Cryptanalysis]]
* [[Cryptography]]
* [[Cybernetics]]
* [[Entropy in thermodynamics and information theory]]
* [[Gambling]]
* [[Intelligence (information gathering)]]
* [[reflection seismology|Seismic exploration]]
{{colend}}

===History===
* [[Ralph Hartley|Hartley, R.V.L.]]
* [[History of information theory]]
* [[Claude Elwood Shannon|Shannon, C.E.]]
* [[Timeline of information theory]]
* [[Hubert Yockey|Yockey, H.P.]]
* [[Andrey Kolmogorov]]

===Theory===
{{div col|colwidth=16em}}
* [[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]]
{{div col end}}

===Concepts===
{{div col|colwidth=16em}}
* [[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 (information theory)|Redundancy]]
* [[Rényi entropy]]
* [[Self-information]]
* [[Unicity distance]]
* [[Variety (cybernetics)|Variety]]
* [[Hamming distance]]
* [[Perplexity]]

{{div col end}}

==References==
{{Reflist}}

== Further reading ==

===The classic work===
{{refbegin}}
* [[Claude Elwood Shannon|Shannon, C.E.]] (1948), "[[A Mathematical Theory of Communication]]", ''Bell System Technical Journal'', 27, pp.&nbsp;379–423 & 623–656, July & October, 1948. [http://math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf PDF.] <br />[https://web.archive.org/web/20150409204946/http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html Notes and other formats.]
* R.V.L. Hartley, [http://www.dotrose.com/etext/90_Miscellaneous/transmission_of_information_1928b.pdf "Transmission of Information"], ''Bell System Technical Journal'', July 1928
* [[Andrey Kolmogorov]] (1968), "[https://www.tandfonline.com/doi/pdf/10.1080/00207166808803030 Three approaches to the quantitative definition of information]" in ''[[International Journal of Computer Mathematics]]'', 2, pp.&nbsp;157–168.
{{refend}}

===Other journal articles===
{{refbegin|}}
* J. L. Kelly Jr., [http://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf Princeton], "A New Interpretation of Information Rate" ''Bell System Technical Journal'', Vol. 35, July 1956, pp.&nbsp;917–26.
* R. Landauer, [https://archive.today/20071016185253/http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=615478 IEEE.org], "Information is Physical" ''Proc. Workshop on Physics and Computation PhysComp'92'' (IEEE Comp. Sci.Press, Los Alamitos, 1993) pp.&nbsp;1–4.
* {{cite journal|last1=Landauer|first1=R.|year=1961|title=Irreversibility and Heat Generation in the Computing Process|url=http://www.research.ibm.com/journal/rd/441/landauerii.pdf|journal=IBM J. Res. Dev.|volume=5|issue=3|pages=183–191|doi=10.1147/rd.53.0183}}
* {{cite arXiv|last1=Timme|first1=Nicholas|last2=Alford|first2=Wesley|last3=Flecker|first3=Benjamin|last4=Beggs|first4=John M.|date=2012|title=Multivariate information measures: an experimentalist's perspective|eprint=1111.6857|class=cs.IT}}
{{refend}}

===Textbooks on information theory===
{{refbegin}}
* Alajaji, F. and Chen, P.N. An Introduction to Single-User Information Theory. Singapore: Springer, 2018. {{isbn|978-981-10-8000-5}}
* Arndt, C.  ''Information Measures, Information and its Description in Science and Engineering'' (Springer Series: Signals and Communication Technology), 2004, {{isbn|978-3-540-40855-0}}
* {{cite book | title = Information Theory | first = Robert B. | last = Ash | location = New York | publisher = Dover Publications, Inc. | orig-year = 1965| year = 1990 | isbn = 0-486-66521-6 | url = https://books.google.com/books?id=ngZhvUfF0UIC&q=intitle:information+intitle:theory+inauthor:ash+conditional+uncertainty&pg=PA16}}
* [[Gallager, R]]. ''Information Theory and Reliable Communication.'' New York: John Wiley and Sons, 1968. {{isbn|0-471-29048-3}}
* Goldman, S. ''Information Theory''. New York: Prentice Hall, 1953. New York: Dover 1968 {{isbn|0-486-62209-6}}, 2005 {{isbn|0-486-44271-3}}
* {{cite book |last1=Cover |first1=Thomas |author-link1=Thomas M. Cover |last2=Thomas |first2=Joy A. |title=Elements of information theory |edition=2nd |location=New York |publisher=[[Wiley-Interscience]] |date=2006 |isbn=0-471-24195-4}}
* [[Csiszar, I]], Korner, J. ''Information Theory: Coding Theorems for Discrete Memoryless Systems''  Akademiai Kiado: 2nd edition, 1997. {{isbn|963-05-7440-3}}
* [[David J. C. MacKay|MacKay, David J. C.]] ''[http://www.inference.phy.cam.ac.uk/mackay/itila/book.html Information Theory, Inference, and Learning Algorithms]'' Cambridge: Cambridge University Press, 2003. {{isbn|0-521-64298-1}}
* Mansuripur, M. ''Introduction to Information Theory''. New York: Prentice Hall, 1987. {{isbn|0-13-484668-0}}
* [[Robert McEliece|McEliece, R]]. ''The Theory of Information and Coding''. Cambridge, 2002.  {{isbn|978-0521831857}}
* [[John R. Pierce|Pierce, JR]].  "An introduction to information theory: symbols, signals and noise".  Dover (2nd Edition). 1961 (reprinted by Dover 1980).
* {{cite book | title = An Introduction to Information Theory | first = Fazlollah M. | last = Reza | author-link=Fazlollah Reza| publisher = Dover Publications, Inc. | location = New York | orig-year = 1961| year = 1994 | isbn = 0-486-68210-2 | url = https://books.google.com/books?id=RtzpRAiX6OgC&q=intitle:%22An+Introduction+to+Information+Theory%22++%22entropy+of+a+simple+source%22&pg=PA8}}
* {{cite book |last1=Shannon |first1=Claude |author-link1=Claude Shannon |last2=Weaver |first2=Warren |author-link2=Warren Weaver |date=1949 |title=The Mathematical Theory of Communication |url=http://monoskop.org/images/b/be/Shannon_Claude_E_Weaver_Warren_The_Mathematical_Theory_of_Communication_1963.pdf |location=[[Urbana, Illinois]] |publisher=[[University of Illinois Press]] |lccn=49-11922 |isbn=0-252-72548-4}}
* Stone, JV.  Chapter 1 of book [https://jamesstone.sites.sheffield.ac.uk/books/information-theory-2nd-edition "Information Theory: A Tutorial Introduction"], University of Sheffield, England, 2014. {{isbn|978-0956372857}}.
* Yeung, RW.  ''[http://iest2.ie.cuhk.edu.hk/~whyeung/book/ A First Course in Information Theory]'' Kluwer Academic/Plenum Publishers, 2002.  {{isbn|0-306-46791-7}}.
* Yeung, RW.  ''[http://iest2.ie.cuhk.edu.hk/~whyeung/book2/ Information Theory and Network Coding]'' Springer 2008, 2002.  {{isbn|978-0-387-79233-0}}
{{refend}}

===Other books===
{{refbegin}}
* Leon Brillouin, ''Science and Information Theory'', Mineola, N.Y.: Dover, [1956, 1962] 2004. {{isbn|0-486-43918-6}}
* {{Cite book |last=Gleick |first=James |author-link=James Gleick |title=The Information: A History, a Theory, a Flood |title-link=The Information: A History, a Theory, a Flood Pantheon Books |year=2011 |isbn=978-1-4000-9623-7 |edition=1st |location=New York |language=en|oclc=607975727|publisher=Pantheon Books}}
* A. I. Khinchin, ''Mathematical Foundations of Information Theory'', New York: Dover, 1957. {{isbn|0-486-60434-9}}
* H. S. Leff and A. F. Rex, Editors, ''Maxwell's Demon: Entropy, Information, Computing'', Princeton University Press, Princeton, New Jersey (1990). {{isbn|0-691-08727-X}}
* [[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}}
* [[Charles Seife]], ''[[Decoding the Universe]]'', Viking, 2006. {{isbn|0-670-03441-X}}
* Jeremy Campbell, ''[[Grammatical Man]]'', Touchstone/Simon & Schuster, 1982, {{isbn|0-671-44062-4}}
* Henri Theil, ''Economics and Information Theory'', Rand McNally & Company - Chicago, 1967.
* Escolano, Suau, Bonev, ''[https://www.springer.com/computer/image+processing/book/978-1-84882-296-2 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. {{ISBN|0-19-923769-7}}
{{refend}}

== External links ==
{{Wikiquote}}
{{Library resources box}}
* {{SpringerEOM |title=Information |id=p/i051040}}
* Lambert F. L. (1999), "[http://jchemed.chem.wisc.edu/Journal/Issues/1999/Oct/abs1385.html Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? Nonsense!]", ''Journal of Chemical Education''
* [http://www.itsoc.org/ IEEE Information Theory Society] and [https://www.itsoc.org/resources/surveys ITSOC Monographs, Surveys, and Reviews] {{Webarchive|url=https://web.archive.org/web/20180612144016/https://www.itsoc.org/resources/surveys |date=2018-06-12 }}

{{Cybernetics}}
{{Informatics}}
{{Compression methods}}
{{Areas of mathematics}}
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{{DEFAULTSORT:Information Theory}}
[[Category:Information theory| ]]
[[Category:Claude Shannon]]
[[Category:Computer-related introductions in 1948]]
[[Category:Cybernetics]]
[[Category:Formal sciences]]
[[Category:History of logic]]
[[Category:History of mathematics]]
[[Category:Information Age]]
[[Category:Data compression]]