信息论基础：第二版英文原版解读

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"Elements of Information Theory 2nd Edition" 是一本由Thomas M. Cover和Joy A. Thomas合著的信息论基础教材，英文原版，由John Wiley & Sons, Inc.出版。《信息论基础》第二版深入浅出地介绍了信息论这一关键的数学理论，它在通信、计算机科学、统计学、机器学习等领域有着广泛的应用。信息论主要研究如何有效地编码、传输和处理信息，以及如何量化信息的含量。本书内容可能涵盖以下几个核心知识点： 1. **信息量与熵**：信息论的基础是信息量的概念，它衡量了一个事件发生的不确定性。熵是描述随机变量不确定性的一个度量，是信息量的期望值。对于离散随机变量，熵定义为各事件概率与其信息量的乘积的加权平均。 2. **信源编码**：讨论如何将来自信源的原始信息转化为适合传输或存储的编码形式。这包括无损编码（如霍夫曼编码）和有损编码（如JPEG图像压缩）。 3. **信道容量**：由香农定理定义，是信道能够无错误传输信息的最大速率。这涉及到信道的带宽、信号噪声比等参数。 4. **信道编码**：通过添加冗余信息来提高信息在有噪声信道中的传输可靠性，如奇偶校验位、循环冗余校验(CRC)和卷积码等。 5. **信息率失真理论**：研究在允许一定程度失真的情况下，如何优化编码以最大化传输速率。 6. **条件熵与互信息**：条件熵描述了在已知某个事件发生的情况下另一个事件的不确定性；互信息则是衡量两个随机变量之间相互依赖程度的量。 7. **网络信息论**：包括多址接入（MAC）和广播信道（BC）的信息论问题，以及马尔科夫决策过程在信息理论中的应用。 8. **率失真函数与源-信道分离定理**：阐述在给定的失真限制下，最优的编码和解码策略，以及在某些条件下，源编码和信道编码可以独立设计的理论。 9. **信源与信道编码定理**：是信息论的核心定理，它们分别给出了达到信源编码和信道编码极限的理论方法。 10. **随机过程在信息论中的应用**：包括高斯白噪声信道、马尔科夫链等，这些在实际通信系统分析中至关重要。这本书不仅对理论进行了深入讲解，还可能包含丰富的实例和习题，帮助读者理解并掌握信息论的基本原理和应用。对于学生和专业人士来说，是一本不可或缺的参考书。

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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