The main reason for the possibility of data compression is the experimental (empirical) law: Real-world sources produce very restricted sets of sequences. How do we model these restrictions? Chapter 10 looks at the first of three compression types that we will consider: variable-length lossless compression.

The platypus is a remarkable animal. It’s characterized by a combination of features that suggest it could belong to any of a few different animal classes. It lays eggs and has a bill like a bird does. It produces milk and has fur like a mammal does. The male platypus carries venomous spikes on its feet, not common for either birds or mammals. Which animal class best suits the platypus? More generally, how should animals be organized into clusters? How many different clusters should there be? Which combinations of features should be considered? What criteria should be used to determine whether animals are similar or dissimilar? What is the benefit of organizing animals into clusters?

The legend goes that Alexander of Macedonia, before embarking on a campaign of world conquest, first visited the Oracle of Delphi, known for her ability to predict the future. Alexander asked the oracle how the campaign would fare, but she replied that such a prediction could not be made with certainty. Unsatisfied, Alexander pressured her until she predicted, “You are invincible, my son!” Then satisfied, Alexander concluded, “Now I have my answer,” and decided to proceed with the campaign.

The question that leads us into the study of different ethical theories concerns the reasons we have to be honest and just in circumstances that invite dishonesty or injustice without risk of disrupting social peace, tarnishing one’s reputation, or losing the goodwill of others. One thought a person who was faced with such circumstances might have is that his happiness is best served in the long run by adhering to the standards of honesty and justice. “The cash is very tempting,” he might say to himself as he looked at the wad of bills in the purse he had just found, “but it would be stupid to take it. The costs and risks involved make it likely to be more trouble than it’s worth.” The ideal that a person who thought along these lines would affirm is that of wisdom in the pursuit of happiness. In ethics, the theory that affirms this ideal is egoism. The popularity of this theory among people unfamiliar with moral philosophy suggests that no other theory has more immediate intuitive appeal. The theory, in addition, has a secure and important place in the history of ethics. Arguably, it is the theory Plato worked out in the Republic to answer Thrasymachus’ challenge to the value of justice. In any case, it certainly had other champions in the ancient world. The most noteworthy of these is the great Hellenistic philosopher Epicurus (341–271 BCE). Its place in modern philosophy is no less prominent. In the early modern period its defenders included such major thinkers as Thomas Hobbes (1588–1679) and Benedict de Spinoza (1632–1677), and it continued to receive strong and important support in the eighteenth and nineteenth centuries. Only in the twentieth century did its vitality begin to wane, although even today it still has active and influential defenders.

In this chapter our goal is to determine the achievable region of the exponent pairs for the type-I and type-II error probabilities. Our strategy is to apply the achievability and (strong) converse bounds from Chapter 14 in conjunction with the large-deviations theory developed in Chapter 15. After characterizing the full tradeoff we will discuss an adaptive setting of hypothesis testing where, instead of committing ahead of time to testing on the basis of n samples, one can decide adaptively whether to request more samples or stop. We will find out that adaptivity greatly increases the region of achievable error exponents and will learn about the sequential probability ratio test (SPRT) of Wald. In the closing sections we will discuss relations to more complicated settings in hypothesis testing: one with composite hypotheses and one with communication constraints.

This chapter reviews the solution to the U(1)A-problem, from a quantitativeperspective. We discuss the ‘t Hooft large-Nclimit of QCD, which removes the axial anomaly, and its reappearance under 1/ Nc corrections. This provides mass to the η’ meson,which is related to the topological susceptibility by the Witten–Veneziano formula. Thissusceptibility is now well-defined on the lattice, based on the index theorem. It has acontrolled continuum limit, which substantiates the Witten–Veneziano formula.

When making a business decision, you can theorize about what happened in the past, is happening now, or will happen in the future. You can then decide what to do based on your best estimates, which will consequently lead to a business result. You might assume that when your estimates are based on patterns you detect in some relevant data, then your estimates will be better, and so your decisions will be better, and so the business result will be better. Indeed, this is often the case. Challenges arise, however, when the patterns in the data are difficult to detect.

A company solicits feedback about its products from its customers. Most customers are pleased with the products, but they don’t bother to respond. Few customers are displeased with the products, but of those few, many of them let the company know about it. The information collected is unbalanced, in the sense that there are fewer observations of pleased customers than of displeased customers. The company already knows the proportions of customers that like or don’t like its products, but wants to better understand what distinguishes a pleased customer from a displeased customer.

Let us return to the problem to which our final criticism of existentialist ethics led. Recall that it concerned whether one’s never taking anyone else’s perspective but one’s own when deliberating about what to do is rationally defensible. If Sidgwick had been successful in his attempt to reconcile utilitarianism with egoism, then he would have shown that rationality required following the Principle of Utility. And because following the principle entails taking a general view of one’s circumstances when deliberating about what to do, he would have then shown that always omitting consideration of others’ perspectives when so deliberating would not be rationally defensible. One cannot, after all, follow it without sometimes considering directly how one’s actions will affect others for good or ill. Sidgwick’s failure to reconcile utilitarianism with egoism does not mean, of course, that always omitting consideration of others’ perspectives is rationally defensible, but it does mean that showing it to be rationally indefensible requires an account of deliberation free of the quandary that stymied Sidgwick’s attempt, namely, the dualism of practical reason. The problem, then, is to find an account of deliberation on which practical reason is unified and its exercise yields principles of right action that require one to consider the perspectives of others.