In Chapter 14 we first define a performance metric giving a full description of the binary hypothesis testing (BHT) problem. A key result in this theory, the Neyman–Pearson lemma, determines the form of the optimal test and at the same time characterizes the given performance metric. We then specialize to the setting of iid observations and consider two types of asymptotics: Stein’s regime (where type-I error is held constant) and Chernoff’s regime (where errors of both types are required to decay exponentially). In this chapter we only discuss Stein's regime and find out that fundamental limit is given by the KL divergence. Subsequent chapters will address the Chernoff's regime.

This chapter provides an extensive discussion of Grand Unified Theories (GUTs) and related subjects. It begins with the SU(5) GUT, its fermion multiplets, and the resulting transitions between leptons and quarks, which enable in particular proton decay. In this context, we discuss the baryon asymmetry in the Universe, as well as possible topological defects dating back to the early Universe, according to the Kibble mechanism, such as domain walls, cosmic strings, or magnetic monopoles. That takes us to a review of Dirac and ‘t Hooft–Polaykov monopoles, Julia–Zee dyons, and the effects named after Callan–Rubakov and Witten. Next we discuss fermion masses in the framework of the GUTs with the gauge groups SU(5) and Spin(10). Then we consider small unified theories (without QCD) with a variety of gauge groups. Finally, we summarize the status and prospects of the GUT approach.

Our discussion of Beauvoir’s theory introduced the possibility of a tyrant who valued dominating others, not as a means to realizing other values, but rather as an ultimate end. Such a figure, you might have thought, appears only in works of fiction as the personification of evil. Yet he is a model of nobility in Nietzsche’s philosophy. Indeed, Beauvoir, in remaking existentialist ethics into a teleological theory, took Nietzsche as an arch opponent whose glorification of man’s will to power, to use his famous trope, nudged existentialism into solipsism. But in this regard she was mistaken. Nietzsche’s extolling of powerful, masterful men as the highest specimens of humanity was neither grounded in nor a springboard for solipsism. Rather it was a distinct echo of Thrasymachus’ views in the first book of the Republic. Like Thrasymachus, Nietzsche separated mankind into the few who were strong and the many who were weak, and like Thrasymachus too Nietzsche had only contempt for the latter and for their appeal to justice as a leveler that raises their fortunes and lowers the fortunes of the former. But unlike Thrasymachus, Nietzsche did not give an argument for his belief that the truly admirable man lives free of the restraints of justice and the other requirements of morality that would keep his desires for self-advancement in check. Thrasymachus’ error was to yield to Socrates’ insistence that he explain his views and submit them to an examination. His error was to put himself on the plane of reason, so to speak, where he was outmaneuvered by Socrates. Nietzsche took a different tack.

First, non-Abelian gauge fields are quantized canonically. The Faddeev–Popov ghost fields implement gauge fixing, then we review the BRST symmetry. Next, we proceed to the lattice regularization and then from Abelian to non-Abelian gauge fields. We stress that the compact lattice functional integral formulation does not require gauge fixing.

We construct mass terms for the Standard Model fermions of the first generation. This includes the neutrino, where we invoke either a dimension-5 term or we add a right-handed neutrino field. We reconsider the CP symmetry, the fate of baryon and lepton numbers, and the quantization of the electric charge. The question of the mass hierarchy takes us to the seesaw mass-by-mixing mechanism. As a peculiarity, we finally revisit such properties in the scenario without colors (Nc=1), which allows leptons and baryons to mix.

In the previous chapter we introduced the concept of variable-length compression and studied its fundamental limits (with and without the prefix-free condition). In some situations, however, one may desire that the output of the compressor always has a fixed length, say, k bits. Unless k is unreasonably large, then, this will require relaxing the losslessness condition. This is the focus of Chapter 11: compression in the presence of (typically vanishingly small) probability of error. It turns out allowing even very small error enables several beautiful effects: The possibility to compress data via matrix multiplication over finite fields (linear compression). The possibility to reduce compression length if side information is available at the decompressor (Slepian–Wolf). The possibility to reduce compression length if access to a compressed representation of side information is available at the decompressor (Ahlswede–Körner–Wyner).

Could a European swallow fly a coconut from the African continent to the British Isles? You can think of bagging as convening a committee of general experts to answer some questions, perhaps questions involving aerodynamics (the study of flying), carpology (the study of seeds and fruits like coconuts), and ornithology (the study of birds like swallows).

In Chapter 19 we apply methods developed in the previous chapters (namely the weak converse and the random/maximal coding achievability) to compute the channel capacity. This latter notion quantifies the maximal amount of (data) bits that can be reliably communicated per single channel use in the limit of using the channel many times. Formalizing the latter statement will require introducing the concept of a communication channel. Then for special kinds of channels (the memoryless and the information-stable ones) we will show that computing the channel capacity reduces to maximizing the (sequence of the) mutual information. This result, known as Shannon’s noisy channel coding theorem, is very special as it relates the value of a (discrete, combinatorial) optimization problem over codebooks to that of a (convex) optimization problem over information measures. It builds a bridge between the abstraction of information measures (Part I) and practical engineering problems.

Eudaimonism was the dominant theory in ancient Greek ethics. The name derives from the Greek word ‘eudaimonia’, which is often translated as ‘happiness’ but is sometimes translated as ‘flourishing.’ Many scholars in fact prefer the latter translation because they believe it better captures the concern of the ancient Greeks with the idea of living well. This preference suggests that a useful way of distinguishing between eudaimonism and egoism is to observe, when formulating their fundamental principles, the distinction between well-being and happiness that we drew in Chapter 2. Accordingly, the fundamental principle of eudaimonism is that the highest good for each person is his or her well-being; the fundamental principle of egoism remains, as before, that the highest good for a person is his or her happiness. Admittedly, this way of distinguishing between the two theories would be theoretically pointless if the determinants of how happy a person was were the same as the determinants of how high a level of well-being the person had achieved. Thus, in particular, when hedonism is the favored theory of well-being, this way of distinguishing between eudaimonism and egoism comes to nothing. It fails in this case to capture any real difference between them. For when hedonism is the favored theory of well-being, determinations of how happy a person is exactly match the determinations of how high a level of well-being a person has achieved.

Both egoism and eudaimonism share an outlook of self-concern. They both identify the perspective from which a person judges what ought to be done as that of someone concerned with how best to promote his own good. On either theory, then, the highest good for a person is that person’s own good, whether this be his own happiness or his own well-being. Hence, on either theory, ethical considerations are understood to have the backing of reason insofar as they help to advance this good.

Chapter 1 introduces the first information measure – Shannon entropy. After studying its standard properties (chain rule, conditioning), we will briefly describe how one could arrive at its definition. We discuss axiomatic characterization, the historical development in statistical mechanics, as well as the underlying combinatorial foundation (“method of types”). We close the chapter with Han’s and Shearer’s inequalities, which both exploit the submodularity of entropy.

Chapter 2 is a study of divergence (also known as information divergence, Kullback–Leibler (KL) divergence, relative entropy), which is the first example of a dissimilarity (information) measure between a pair of distributions P and Q. Defining KL divergence and its conditional version in full generality requires some measure-theoretic acrobatics (Radon–Nikodym derivatives and Markov kernels) that we spend some time on. (We stress again that all this abstraction can be ignored if one is willing to work only with finite or countably infinite alphabets.) Besides definitions we prove the “main inequality” showing that KL divergence is non-negative. Coupled with the chain rule for divergence, this inequality implies the data-processing inequality, which is arguably the central pillar of information theory and this book. We conclude the chapter by studying the local behavior of divergence when P and Q are close. In the special case when P and Q belong to a parametric family, we will see that divergence is locally quadratic, with Hessian being the Fisher information, explaining the fundamental role of the latter in classical statistics.

Chiral perturbation theory is the systematic low-energy effective theory of QCD, in terms of low-energy parameters and pseudo-Nambu–Goldstone boson fields representing pions, kaons, and the η. We discuss their masses in leading order, and the corresponding electromagnetic corrections, where we arrive at Dashen’s theorem. We show how this low-energy scheme even encompasses nucleons, and how QCD provides corrections to the weak gauge boson masses. In that context, we comment on a technicolor extension and on the hypothesis of minimal flavor violation, which is described by spurions.

This enthusiastic introduction to the fundamentals of information theory builds from classical Shannon theory through to modern applications in statistical learning, equipping students with a uniquely well-rounded and rigorous foundation for further study. The book introduces core topics such as data compression, channel coding, and rate-distortion theory using a unique finite blocklength approach. With over 210 end-of-part exercises and numerous examples, students are introduced to contemporary applications in statistics, machine learning, and modern communication theory. This textbook presents information-theoretic methods with applications in statistical learning and computer science, such as f-divergences, PAC-Bayes and variational principle, Kolmogorov’s metric entropy, strong data-processing inequalities, and entropic upper bounds for statistical estimation. Accompanied by additional stand-alone chapters on more specialized topics in information theory, this is the ideal introductory textbook for senior undergraduate and graduate students in electrical engineering, statistics, and computer science.

We outline the main concepts of the Standard Model, illustratively describing its central features and some open questions, as a preparation for the following chapters.