信息论基础：Cover & Thomas的经典教材第二版

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"Elements of Information Theory" 是一本由Thomas M. Cover和Joy A. Thomas合著的经典信息论教材，第二版延续了其清晰、启发思考的教学风格，涵盖了数学、物理、统计学和信息理论的综合知识。信息论是研究信息的量化、传输和处理的学科，起源于20世纪40年代，由克劳德·香农(Claude Shannon)创立。本书作为信息论的基础教材，深入浅出地讲解了这一领域的核心概念和原理，适合学生和专业人士学习。主要内容包括： 1. 熵（Entropy）：熵是信息论中的基本概念，用来衡量一个随机变量的不确定性。在信息论中，熵通常用比特(bit)来度量，表示消息平均所需的最小信息量。例如，二进制系统中，事件发生的概率越均匀，熵越大，表示信息量越大。 2. 互信息（Mutual Information）：互信息描述了两个随机变量之间的关联程度，是衡量一个变量通过另一个变量获得的信息量。在通信中，它有助于理解信号传输的有效性。 3. 信道容量（Channel Capacity）：香农第一定理阐述了无错误通信的信道容量，这是在给定的噪声水平下，一个通信信道能够无错误传输的最大数据速率。计算信道容量的关键在于找到最佳的输入分布。 4. 编码理论（Coding Theory）：信息论中的编码理论研究如何有效地编码信息以提高传输效率和鲁棒性，比如纠错码和信源编码。 5. 信息压缩（Data Compression）：通过对数据进行编码，可以减少存储和传输所需的空间，如哈夫曼编码(Huffman Coding)和算术编码(Arithmetic Coding)。 6. 信道编码（Channel Coding）：通过在原始信息中添加冗余，可以检测并纠正传输过程中的错误，如线性分组码和卷积码。 7. 噪声与信噪比（Noise and Signal-to-Noise Ratio, SNR）：在通信系统中，噪声是不可避免的干扰，信噪比是信号功率与噪声功率之比，是衡量通信质量的重要指标。 8. 最大似然估计（Maximum Likelihood Estimation）：在统计推断中，最大似然估计是一种常用的参数估计方法，通过最大化观测数据出现的概率来估计未知参数。 9. 信息理论与机器学习的关系：信息论的概念也被广泛应用于现代机器学习领域，如特征选择、模型复杂度控制以及生成对抗网络(GANs)的设计。本书不仅涵盖了上述基础知识，还可能深入到更高级的主题，如多元信息论、网络信息论、量子信息论等。通过阅读和学习，读者将对信息处理的基本原理有深入的理解，并能应用这些理论解决实际问题。

PREFACE TO THE

SECOND EDITION

In the years since the publication of the ﬁrst edition, there were many

aspects of the book that we wished to improve, to rearrange, or to expand,

but the constraints of reprinting would not allow us to make those changes

between printings. In the new edition, we now get a chance to make some

of these changes, to add problems, and to discuss some topics that we had

omitted from the ﬁrst edition.

The key changes include a reorganization of the chapters to make

the book easier to teach, and the addition of more than two hundred

new problems. We have added material on universal portfolios, universal

source coding, Gaussian feedback capacity, network information theory,

and developed the duality of data compression and channel capacity. A

new chapter has been added and many proofs have been simpliﬁed. We

have also updated the references and historical notes.

The material in this book can be taught in a two-quarter sequence. The

ﬁrst quarter might cover Chapters 1 to 9, which includes the asymptotic

equipartition property, data compression, and channel capacity, culminat-

ing in the capacity of the Gaussian channel. The second quarter could

cover the remaining chapters, including rate distortion, the method of

types, Kolmogorov complexity, network information theory, universal

source coding, and portfolio theory. If only one semester is available, we

would add rate distortion and a single lecture each on Kolmogorov com-

plexity and network information theory to the ﬁrst semester. A web site,

http://www.elementsoﬁnformationtheory.com, provides links to additional

material and solutions to selected problems.

In the years since the ﬁrst edition of the book, information theory

celebrated its 50th birthday (the 50th anniversary of Shannon’s original

paper that started the ﬁeld), and ideas from information theory have been

applied to many problems of science and technology, including bioin-

formatics, web search, wireless communication, video compression, and

PREFACE TO THE

FIRST EDITION

This is intended to be a simple and accessible book on information theory.

As Einstein said, “Everything should be made as simple as possible, but no

simpler.” Although we have not veriﬁed the quote (ﬁrst found in a fortune

cookie), this point of view drives our development throughout the book.

There are a few key ideas and techniques that, when mastered, make the

subject appear simple and provide great intuition on new questions.

This book has arisen from over ten years of lectures in a two-quarter

sequence of a senior and ﬁrst-year graduate-level course in information

theory, and is intended as an introduction to information theory for stu-

dents of communication theory, computer science, and statistics.

There are two points to be made about the simplicities inherent in infor-

mation theory. First, certain quantities like entropy and mutual information

arise as the answers to fundamental questions. For example, entropy is

the minimum descriptive complexity of a random variable, and mutual

information is the communication rate in the presence of noise. Also,

as we shall point out, mutual information corresponds to the increase in

the doubling rate of wealth given side information. Second, the answers

to information theoretic questions have a natural algebraic structure. For

example, there is a chain rule for entropies, and entropy and mutual infor-

mation are related. Thus the answers to problems in data compression

and communication admit extensive interpretation. We all know the feel-

ing that follows when one investigates a problem, goes through a large

amount of algebra, and ﬁnally investigates the answer to ﬁnd that the

entire problem is illuminated not by the analysis but by the inspection of

the answer. Perhaps the outstanding examples of this in physics are New-

ton’s laws and Schr

odinger’s wave equation. Who could have foreseen the

awesome philosophical interpretations of Schr

odinger’s wave equation?

In the text we often investigate properties of the answer before we look

at the question. For example, in Chapter 2, we deﬁne entropy, relative

entropy, and mutual information and study the relationships and a few

xvii

xviii PREFACE TO THE FIRST EDITION

interpretations of them, showing how the answers ﬁt together in various

ways. Along the way we speculate on the meaning of the second law of

thermodynamics. Does entropy always increase? The answer is yes and

no. This is the sort of result that should please experts in the area but

might be overlooked as standard by the novice.

In fact, that brings up a point that often occurs in teaching. It is fun

to ﬁnd new proofs or slightly new results that no one else knows. When

one presents these ideas along with the established material in class, the

response is “sure, sure, sure.” But the excitement of teaching the material

is greatly enhanced. Thus we have derived great pleasure from investigat-

ing a number of new ideas in this textbook.

Examples of some of the new material in this text include the chapter

on the relationship of information theory to gambling, the work on the uni-

versality of the second law of thermodynamics in the context of Markov

chains, the joint typicality proofs of the channel capacity theorem, the

competitive optimality of Huffman codes, and the proof of Burg’s theorem

on maximum entropy spectral density estimation. Also, the chapter on

Kolmogorov complexity has no counterpart in other information theory

texts. We have also taken delight in relating Fisher information, mutual

information, the central limit theorem, and the Brunn–Minkowski and

entropy power inequalities. To our surprise, many of the classical results

on determinant inequalities are most easily proved using information the-

oretic inequalities.

Even though the ﬁeld of information theory has grown considerably

since Shannon’s original paper, we have strived to emphasize its coher-

ence. While it is clear that Shannon was motivated by problems in commu-

nication theory when he developed information theory, we treat informa-

tion theory as a ﬁeld of its own with applications to communication theory

and statistics. We were drawn to the ﬁeld of information theory from

backgrounds in communication theory, probability theory, and statistics,

because of the apparent impossibility of capturing the intangible concept

of information.

Since most of the results in the book are given as theorems and proofs,

we expect the elegance of the results to speak for themselves. In many

cases we actually describe the properties of the solutions before the prob-

lems. Again, the properties are interesting in themselves and provide a

natural rhythm for the proofs that follow.

One innovation in the presentation is our use of long chains of inequal-

ities with no intervening text followed immediately by the explanations.

By the time the reader comes to many of these proofs, we expect that he

or she will be able to follow most of these steps without any explanation

and will be able to pick out the needed explanations. These chains of

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