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{{distinguish|Information science}}
{{distinguish|Information science}}
{{Information theory}}
{{Information theory}}
'''Information theory''' studies the [[quantification (science)|quantification]], [[computer data storage|storage]], and [[telecommunication|communication]] of [[information]]. It was originally proposed by [[Claude Shannon]] in 1948 to find fundamental limits on [[signal processing]] and communication operations such as [[data compression]], in a landmark paper titled "[[A Mathematical Theory of Communication]]". Its impact has been crucial to the success of the [[Voyager program|Voyager]] missions to deep space, the invention of the [[compact disc]], the feasibility of mobile phones, the development of the Internet, the study of [[linguistics]] and of human perception, the understanding of [[black hole]]s, and numerous other fields.
'''Information theory''' studies the [[quantification (science)|quantification]], [[computer data storage|storage]], and [[telecommunication|communication]] of [[information]]. It was originally proposed by [[Claude Shannon]] in 1948 to find fundamental limits on [[signal processing]] and communication operations such as [[data compression]], in a landmark paper titled "[[A Mathematical Theory of Communication]]". Its impact has been crucial to the success of the [[Voyager program|Voyager]] missions to deep space, the invention of the [[compact disc]], the feasibility of mobile phones, the development of the Internet, the study of [[linguistics]] and of human perception, the understanding of [[black hole]]s, and numerous other fields.
Information theory studies the quantification, storage, and communication of information. It was originally proposed by Claude Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper titled "A Mathematical Theory of Communication". Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields.
Information theory studies the quantification, storage, and communication of information. It was originally proposed by Claude Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper titled "A Mathematical Theory of Communication". Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields.
'''信息论'''研究的是信息的量化、存储与传播。信息论最初是由[[Claude Shannon]]在1948年的一篇题为"[[A Mathematical Theory of Communication]]"的论文中提出的，其目的是找到信号处理和通信操作（如数据压缩）的基本限制。信息论对于旅行者号深空探测任务的成功、光盘的发明、移动电话的可行性、互联网的发展、语言学和人类感知的研究、对黑洞的理解以及许多其他领域的研究都是至关重要的。
'''信息论'''研究的是信息的量化、存储与传播。信息论最初是由'''克劳德·香农 Claude Shannon'''在1948年的一篇题为"《一种通信的数学理论（A Mathematical Theory of Communication）》"的里程碑式论文中提出的，其目的是找到信号处理和通信操作（如数据压缩）的基本限制。信息论对于旅行者号深空探测任务的成功、光盘的发明、移动电话的可行性、互联网的发展、语言学和人类感知的研究、对黑洞的理解以及许多其他领域的研究都是至关重要的。
The field is at the intersection of mathematics, [[statistics]], computer science, physics, [[Neuroscience|neurobiology]], [[information engineering (field)|information engineering]], and electrical engineering. The theory has also found applications in other areas, including [[statistical inference]], [[natural language processing]], [[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> [[human vision]],<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> 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|url=|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}}</ref> of molecular codes ([[bioinformatics]]), [[model selection]] in statistics,<ref>Burnham, K. P. and Anderson D. R. (2002) ''Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Second Edition'' (Springer Science, New York) {{ISBN|978-0-387-95364-9}}.</ref> [[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}}</ref> [[quantum computing]], linguistics, [[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]], and [[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|archiveurl=https://web.archive.org/web/20110723045720/http://aicanderson2.home.comcast.net/~aicanderson2/home.pdf|archivedate=July 23, 2011|url-status=dead|accessdate=2010-06-23}}
</ref> Important sub-fields of information theory include [[source coding]], [[algorithmic complexity theory]], [[algorithmic information theory]], and [[information-theoretic security]].
The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, information engineering, and electrical engineering. The theory has also found applications in other areas, including statistical inference, natural language processing, cryptography, neurobiology, human vision, the evolution and function of molecular codes (bioinformatics), model selection in statistics, thermal physics, quantum computing, linguistics, plagiarism detection, pattern recognition, and anomaly detection.
The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, information engineering, and electrical engineering. The theory has also found applications in other areas, including statistical inference, natural language processing, cryptography, neurobiology, human vision, the evolution and function of molecular codes (bioinformatics), model selection in statistics, thermal physics, quantum computing, linguistics, plagiarism detection, pattern recognition, and anomaly detection.<ref>
该领域是数学、统计学、计算机科学、物理学、神经生物学、信息工程和电气工程的交叉学科。这一理论也在其他领域得到了应用，比如推论统计学、自然语言处理、密码学、神经生物学、人类视觉、分子编码的进化和功能(生物信息学)、统计学中的模型选择、热物理学、量子计算、语言学、剽窃检测、模式识别和异常检测。
该领域是数学、统计学、计算机科学、物理学、神经生物学、信息工程和电气工程的交叉学科。这一理论也在其他领域得到了应用，比如推论统计学、自然语言处理、密码学、神经生物学、人类视觉、分子编码的进化和功能(生物信息学)、统计学中的模型选择、热物理学、量子计算、语言学、剽窃检测、模式识别和异常检测。
Important sub-fields of information theory include [[source coding]], [[algorithmic complexity theory]], [[algorithmic information theory]], [[information-theoretic security]], [[Grey system theory]] and measures of information.
Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory, information-theoretic security, Grey system theory and measures of information.
信息论的重要分支包括信源编码、算法复杂性理论、算法信息论、信息理论安全性、灰色系统理论和信息度量。
Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s and JPEGs), and channel coding (e.g. for DSL). Information theory is used in information retrieval, intelligence gathering, gambling, and even in musical composition.
Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files), lossy data compression (e.g. MP3s and JPEGs), and channel coding (e.g. for DSL). Information theory is used in information retrieval, intelligence gathering, gambling, and even in musical composition.
信息论在应用领域的基本课题包括无损数据压缩(例如:ZIP压缩文件)、有损数据压缩(例如:Mp3和jpeg格式) ，以及频道编码(例如:DSL)。信息论在信息检索、情报收集、赌博，甚至在音乐创作中也有应用。
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 flip]] (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a {{dice}} (with six equally likely outcomes). Some other important measures in information theory are [[mutual information]], channel capacity, [[error exponent]]s, and [[relative entropy]].
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 flip]] (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a {{dice}} (with six equally likely outcomes). Some other important measures in information theory are [[mutual information]], channel capacity, [[error exponent]]s, and [[relative entropy]].
A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy.
A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a dice (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy.
信息论中的一个关键度量是熵。熵量化了一个随机变量的值或者一个随机过程的结果所包含的不确定性。例如，识别一次公平抛硬币的结果(有两个同样可能的结果)所提供的信息(较低的熵)少于识别抛一次骰子的结果(有六个同样可能的结果)。信息论中的其他一些重要指标有：互信息、信道容量、误差指数和相对熵。
==概览==
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 made concrete in 1948 by Claude Shannon in his paper "A Mathematical Theory of Communication", in which "information" is thought of as a set of possible messages, where the goal is to send these messages over a noisy channel, and then 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" />
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 made concrete in 1948 by Claude Shannon in his paper "A Mathematical Theory of Communication", in which "information" is thought of as a set of possible messages, where the goal is to send these messages over a noisy channel, and then 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" />
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 made concrete in 1948 by Claude Shannon in his paper "A Mathematical Theory of Communication", in which "information" is thought of as a set of possible messages, where the goal is to send these messages over a noisy channel, and then 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.
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 made concrete in 1948 by Claude Shannon in his paper "A Mathematical Theory of Communication", in which "information" is thought of as a set of possible messages, where the goal is to send these messages over a noisy channel, and then 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.
信息论主要研究信息的传递、处理、提取和利用。抽象地说，信息可以作为不确定性的解决方案。1948年，克劳德·香农在他的论文《一种通信的数学理论》中将这个抽象的概念具体化，在这篇论文中“信息”被认为是一组可能的信号，这些信号在通过带有噪声的信道发送后，接收者能在信道噪声的影响下以较低的错误概率来重构这些信号。香农的主要结论，有噪信道编码定理，表明在信道使用的许多限制情况下，渐近可达到信息传输速率等于的信道容量，一个仅仅依赖于信息发送所经过的信道本身的统计量。（译注：当信道的信息传输率不超过信道容量时，采用合适的编码方法可以实现任意高的传输可靠性，但若信息传输率超过了信道容量，就不可能实现可靠的传输。）
Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half century or more: adaptive systems, anticipatory systems, artificial intelligence, complex systems, complexity science, cybernetics, informatics, machine learning, along with systems sciences of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.
Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half century or more: adaptive systems, anticipatory systems, artificial intelligence, complex systems, complexity science, cybernetics, informatics, machine learning, along with systems sciences of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.
信息论与一系列纯科学和应用科学密切相关。在过去半个世纪甚至更久的时间里，在全球范围内已经有各种各样的学科理论被研究和化归为工程实践，比如在自适应系统，预期系统，人工智能，复杂系统，复杂性科学，控制论，信息学，机器学习，以及系统科学。信息论是一个广博而深遂的数学理论，也具有广泛而深入的应用，其中编码理论是至关重要的领域。
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.
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.
编码理论与寻找明确的方法（编码）有关，用于提高效率和将有噪信道上传输的数据错误率降低到接近信道容量。这些编码可大致分为数据压缩编码(信源编码)和纠错(信道编码)技术。对于纠错技术，香农证明了理论极限很多年后才有人找到了真正实现了理论最优的方法。
A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban (unit) for a historical application.
A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban (unit) for a historical application.
第三类信息论代码是密码算法(包括代码和密码)。编码理论和信息论的概念、方法和结果在密码学和密码分析中得到了广泛的应用。有关历史应用，请参阅文章禁令（单位）。
第三类信息论代码是密码算法(包括密文和密码)。编码理论和信息论的概念、方法和结果在密码学和密码分析中得到了广泛的应用。
==历史背景==
{{Main|History of information theory}}
{{Main|History of information theory}}
The landmark event that established the discipline of information theory and brought it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948.
The landmark event that established the discipline of information theory and brought it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948.
1948年7月和10月，克劳德·E·香农在《贝尔系统技术期刊》上发表了经典论文：《一种通信的数学理论》，这就是建立信息论学科并立即引起全世界关注的里程碑事件。
Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation (recalling Boltzmann's constant), where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as , where S was the number of possible symbols, and n the number of symbols in a transmission. The unit of information was therefore the decimal digit, which has since sometimes been called the hartley in his honor as a unit or scale or measure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.
Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation (recalling Boltzmann's constant), where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other, thus quantifying information as , where S was the number of possible symbols, and n the number of symbols in a transmission. The unit of information was therefore the decimal digit, which has since sometimes been called the hartley in his honor as a unit or scale or measure of information. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers.
在此之前，贝尔实验室已经提出了有限的信息论思想，所有这些理论都隐性地假设了概率均等的事件。Harry Nyquist 在1924年发表的论文《集中影响电报速率的因素（Certain Factors Affecting Telegraph Speed）》中包含一个理论章节，量化了“情报”和通信系统可以传输的“线路速度”，并给出了关系式 {{math|1=''W'' = ''K'' log ''m''}} (参考玻尔兹曼常数) ，其中 ''W'' 是情报传输的速度， ''m'' 是每个时间步长可以选择的不同电压电平数，''K'' 是常数。Ralph Hartley 在1928年发表的论文《信息的传输（ Transmission of Information）》中，将单词信息作为一个可测量的量，以此反映接收者区分一系列符号的能力，从而将信息量化为 {{math|1=''H'' = log ''S''<sup>''n''</sup> = ''n'' log ''S''}}，其中 ''S'' 是可以使用的符号的数量，''n'' 是传输中符号的数量。因此信息的单位就是十进制数字，为了表示对他的尊敬，这个单位有时被称为 Hartley，作为信息的单位、尺度或度量。1940年，图灵在二战时期破解德国的“迷”密码（Enigma ciphers）的统计分析中使用了类似的思想。
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]]''.
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]]''.
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.
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.
信息论背后的许多数学理论（包括不同概率的事件）都是由路德维希·玻尔兹曼和约西亚·威拉德·吉布斯为热力学领域开发出来的。
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 that
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 that
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 that
香农的那篇革命性的、开创性的论文，于1944年的年底便已基本在贝尔实验室完成。在这论文里，香农将通信看作一个统计学过程，首次提出了通信的量化模型，并以此为基础推导出了信息论。论文开篇便提出了一下论断：
:"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."
:"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."
"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."
"The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."
“通信的基本问题是在一点上精确地或近似地再现在另一点上选择的信息”
With it came the ideas of
With it came the ideas of
With it came the ideas of
With it came the ideas of
与此相关的一些想法包括：
* the information entropy and [[redundancy (information theory)|redundancy]] of a source, and its relevance through the [[source coding theorem]];
* the information entropy and [[redundancy (information theory)|redundancy]] of a source, and its relevance through the [[source coding theorem]];
* 信息熵和信源冗余，以及信源编码定理；
* 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 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;
* 互信息，有噪信道的信道容量，包括无损通信的证明，和有噪信道编码定理；
* the practical result of the [[Shannon–Hartley law]] for the channel capacity of a [[Gaussian channel]]; as well as
* the practical result of the [[Shannon–Hartley law]] for the channel capacity of a [[Gaussian channel]]; as well as
* 香农-哈特利定律应用于高斯信道的信道容量的结果，以及
* the [[bit]]—a new way of seeing the most fundamental unit of information.
* the [[bit]]—a new way of seeing the most fundamental unit of information.
* 比特——一种新的度量信息的最基本单位
==信息的度量==
==Quantities of information==
{{Main|Quantities of information}}
{{Main|Quantities of information}}
Information theory is based on [[probability theory]] and statistics. Information theory often concerns itself with measures of information of the distributions associated with random variables. Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, a measure of information in common between two random variables. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy [[Communication channel|channel]] in the limit of long block lengths, when the channel statistics are determined by the joint distribution.
Information theory is based on [[probability theory]] and statistics. Information theory often concerns itself with measures of information of the distributions associated with random variables. Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, a measure of information in common between two random variables. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy [[Communication channel|channel]] in the limit of long block lengths, when the channel statistics are determined by the joint distribution.
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 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.
信息论基于概率论和统计学，其中经常涉及衡量随机变量的分布的信息。信息论中重要的信息量有：熵（单个随机变量中信息的度量）和互信息（两个随机变量之间的信息的度量）。熵是随机变量的概率分布的一个属性，它限制了从给定分布中独立采样得到的数据的压缩率。互信息是两个随机变量的联合概率分布的一个属性，是当信道的统计量由联合分布确定时，在长块长度的限制下，通过有噪信道的可靠通信的最大速率。
The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm.
The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm.
在下列公式中，对数底数的选择决定了信息熵的单位。信息的常见单位是比特（基于二进制对数）。其他单位包括 nat（自然对数）和十进制数字（常用对数）。
In what follows, an expression of the form is considered by convention to be equal to zero whenever . This is justified because <math>\lim_{p \rightarrow 0+} p \log p = 0</math> for any logarithmic base.
In what follows, an expression of the form is considered by convention to be equal to zero whenever . This is justified because <math>\lim_{p \rightarrow 0+} p \log p = 0</math> for any logarithmic base.
下文中，按惯例将 {{math|''p'' = 0}} 时的表达式{{math|''p'' log ''p''}}的值视为等于零，因为<math>\lim_{p \rightarrow 0+} p \log p = 0</math>适用于任何对数底。
信源的熵
===信源的熵===
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
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