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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, 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。  

</ref> 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.

 

 

 

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.

信息论基本主题的应用包括无损数据压缩(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)。信息论应用于信息检索、情报收集、赌博，甚至在音乐创作中也有应用。

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.

 

 

 

 

 

 

 

 

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.

 

 

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.

 

 

 

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，用于提高效率和将噪声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。在后一种技术中，花了很多年才证明Shannon的工作是可行的。

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。

{{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.

 

 

 

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，所有这些理论都隐含地假设了概率均等的事件。Harry Nyquist 在1924年发表的论文”Certain Factors Affecting Telegraph Speed”中包含一个理论部分，量化了“智能”和通信系统可以传输的“线路速度”，并给出了关系式(检索Boltzmann常数) ，其中 w 是智能传输的速度，m 是每个时间步长可以选择的不同电压电平的数，k 是常数。Ralph Hartley在1928年发表的论文” Transmission of Information”中，将单词信息作为一个可测量的量，以此反映接收器区分一系列符号的能力，从而将信息量化，其中 s 是可能符号的数量，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，Alan Turing在对德国二战时期破解迷密码（Enigma ciphers）的统计分析中使用了类似的思想。

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''^''n'' = ''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.

 

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

:"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."

 

 

 

 

With it came the ideas of

With it came the ideas of

With it came the ideas of

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 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 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 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.

* the [[bit]]—a new way of seeing the most fundamental unit of information.

 

==信息的度量==

 

 

 

 

 

 

==Quantities of information==

 

==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.

 

 

 

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.

 

 

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，按惯例将形式的表达式视为等于零。这是合理的，因为<math>\lim_{p \rightarrow 0+} p \log p = 0</math>适用于任何对数底。

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