Introduction

We have abandoned many of the goals of the early writers on induction. Probability has told us nothing about how to find interesting generalizations and theories, and, although Carnap and others had hoped otherwise, it has told us nothing about how to measure the support for generalizations other than approximate statistical hypotheses. Much of uncertain inference has yet to be characterized in the terms we have used for statistical inference. Let us take a look at where we have arrived so far.

Objectivity

Our overriding concern has been with objectivity. We have looked on logic as a standard of rational argument: Given evidence (premises), the validity (degree of entailment) of a conclusion should be determined on logical grounds alone. Given that the Hawks will win or the Tigers will win, and that the Tigers will not win, it follows that the Hawks will win. Given that 10% of a large sample of trout from Lake Seneca have shown traces of mercury, and that we have no grounds for impugning the fairness of the sample, it follows with a high degree of validity that between 8% and 12% of the trout in the lake contain traces of mercury.

The parallel is stretched only at the point where we include among the premises “no grounds for impugning. …” It is this that is unpacked into a claim about our whole body of knowledge, and embodied in the constraints discussed in the last three chapters under the heading of “sharpening.”

The system described in this book retrieves and analyzes data each night and employs a substantial amount of computational sophistication to determine the most profitable bets to make. It isn't something you are going to try at home, kiddies.

However, in this section I'll provide some hints on how you can make your trip to the fronton as profitable as possible. By combing the results of our Monte Carlo simulations and expected payoff model, I've constructed tables giving the expected payoff for each bet, under the assumption that all players are equally skillful. This is very useful information to have if you are not equipped to make your own judgments as to who is the best player, although we also provide tips as to how to assess player skills. By following my advice, you will avoid criminally stupid bets like the 6–8–7 trifecta.

But first a word of caution is in order. There are three primary types of gamblers:

• Those who gamble to make money – If you are in this category, you are likely a sick individual and need help. My recommendation instead would be that you take your money and invest in a good mutual fund. In particular, the Vanguard Primecap fund has done right well for me over the past few years.

1. One theme running through this book is how hard we had to work in order to make even a small profit. As the saying goes, “gambling is a hard way to make easy money.”

2. […]

Introduction

We are now in a position to reap the benefits of the formal work of the preceding two chapters. The key to uncertain inference lies, as we have suspected all along, in probability. In Chapter 9, we examined a certain formal interpretation of probability, dubbed evidential probability, as embodying a notion of partial proof. Probability, on this view, is an interval-valued function. Its domain is a combination of elementary evidence and general background knowledge paired with a statement of our language whose probability concerns us, and its range is of [0, 1]. It is objective. What this means is that if two agents share the same evidence and the same background knowledge, they will assign the same (interval) probabilities to the statements of their language. If they share an acceptance level 1 – α for practical certainty, they will accept the same practical certainties.

It may be that no two people share the same background knowledge and the same evidence. But in many situations we come close. As scientists, we tend to share each other's data. Cooked data is sufficient to cause expulsion from the ranks of scientists. (This is not the same as data containing mistakes; one of the virtues of the system developed here is that no data need be regarded as sacrosanct.) With regard to background knowledge, if we disagree, we can examine the evidence at a higher level: is the item in question highly probable, given that evidence and our common background knowledge at that level?

There are a number of epistemological questions raised by this approach, and some of them will be dealt with in Chapter 12.

Mathematical modeling is a subject best appreciated by doing. The trick is finding an interesting type of prediction to make or question to study, and then identifying sufficient data to build a reasonable model upon. Even if you are not a computer programmer, spread-sheet programs such as Microsoft Excel can provide an excellent environment in which to experiment with mathematical models.

In this section, I pose several interesting questions to which the modeling techniques presented in this book may be applicable. To provide starting points, I include links to existing studies and data sets on the WWW. Web links are extremely perishable, so treat these only as an introduction. Any good search engine like www.google.com should help you find better sources after a few minutes' toil. Happy modeling!

Gambling

• Lottery numbers – How random are lottery numbers? Do certain numbers in certain states come up more often than would be expected by chance? Can you predict which lotto combinations are typically underbet, meaning that they minimize the likelihood that you must share the pot with someone else if you win?. How large must pool size grow in a given progressive lottery to yield a positive expected value for each ticket bought?

1. Plenty of lottery records are available on the WWW if you look hard enough. Log on to http://www.lottonet.com/ for several year's historical data from several state lotteries. Minnesota does a particularly good job, making its historical numbers available at http://www.lottery.state.mn.us.

• Horse racing – Many of the ideas employed in our jai alai system are directly applicable to horse racing. […]

In this chapter, we discuss a number of interrelated subjects: how to use the Internet to get additional help with MATLAB and to find MATLAB programs for certain specific applications, how to disseminate MATLAB programs over the Internet, and how to use MATLAB to prepare documents for posting on the World Wide Web.

MATLAB Help on the Internet

For answers to a variety of questions about MATLAB, it pays to visit the web site for The MathWorks,

(In MATLAB 6, the Web menu on the Desktop menu bar can take you there automatically.) Since files on this site are moved around periodically, we won't tell you precisely what is located where, but we will point out a few things to look for. First, you can find complete documentation sets for MATLAB and all the toolboxes. This is particularly useful if you didn't install all the documentation locally in order to save space. Second, there are lists of frequently asked questions about MATLAB, bug reports and bug fixes, etc. Third, there is an index of MATLAB-based books (including this one), with descriptions and ordering information. And finally, there are libraries of M-files, developed both by The MathWorks and by various MATLAB users, which you can download for free. These are especially useful if you need to do a standard sort of calculation for which there are established algorithms but for which MATLAB has no built-in M-file; in all probability, someone has written an M-file for it and made it available.

Cryptography is concerned with the construction of schemes that should be able to withstand any abuse. Such schemes are constructed so as to maintain a desired functionality, even under malicious attempts aimed at making them deviate from their prescribed functionality.

The design of cryptographic schemes is a very difficult task. One cannot rely on intuitions regarding the typical state of the environment in which a system will operate. For sure, an adversary attacking the system will try to manipulate the environment into untypical states. Nor can one be content with countermeasures designed to withstand specific attacks, because the adversary (who will act after the design of the system has been completed) will try to attack the schemes in ways that typically will be different from the ones the designer envisioned. Although the validity of the foregoing assertions seems self-evident, still some people hope that, in practice, ignoring these tautologies will not result in actual damage. Experience shows that such hopes are rarely met; cryptographic schemes based on make-believe are broken, typically sooner rather than later.

In view of the foregoing, we believe that it makes little sense to make assumptions regarding the specific strategy that an adversary may use. The only assumptions that can be justified refer to the computational abilities of the adversary.

In this chapter we describe SIMULINK, a MATLAB accessory for simulating dynamical processes, and GUIDE, a built-in tool for creating your own graphical user interfaces. These brief introductions are not comprehensive, but together with the online documentation they should be enough to get you started.

SIMULINK

If you want to learn about SIMULINK in depth, you can read the massive PDF document SIMULINK: Dynamic System Simulation for MATLAB that comes with the software. Here we give a brief introduction for the casual user who wants to get going with SIMULINK quickly. You start SIMULINK by doubleclicking on SIMULINK in the Launch Pad, by clicking on the SIMULINK button on the MATLAB Desktop tool bar, or simply by typing simulink in the Command Window. This opens the SIMULINK library window, which is shown for UNIX systems in Figure 8-1. On Windows systems, you see instead the SIMULINK Library Browser, shown in Figure 8-2.

To begin to use SIMULINK, click New: Model from the File menu. This opens a blank model window. You create a SIMULINK model by copying units, called blocks, from the various SIMULINK libraries into the model window. We will explain how to use this procedure to model the homogeneous linear ordinary differential equation u″ + 2u′ + 5u = 0, which represents a damped harmonic oscillator.

First we have to figure out how to represent the equation in a way that SIMULINK can understand. One way to do this is as follows.

That statement encapsulates the view of The Math Works, Inc., the developer of MATLAB®. MATLAB 6 is an ambitious program. It contains hundreds of commands to do mathematics. You can use it to graph functions, solve equations, perform statistical tests, and do much more. It is a high-level programming language that can communicate with its cousins, e.g., FORTRAN and C. You can produce sound and animate graphics. You can do simulations and modeling (especially if you have access not just to basic MATLAB but also to its accessory SIMULINK®). You can prepare materials for export to the World Wide Web. In addition, you can use MATLAB, in conjunction with the word processing and desktop publishing features of Microsoft Word®, to combine mathematical computations with text and graphics to produce a polished, integrated, and interactive document.

A program this sophisticated contains many features and options. There are literally hundreds of useful commands at your disposal. The MATLAB help documentation contains thousands of entries. The standard references, whether the MathWorks User's Guide for the product, or any of our competitors, contain myriad tables describing an endless stream of commands, options, and features that the user might be expected to learn or access.

MATLAB is more than a fancy calculator; it is an extremely useful and versatile tool.

Support

As Carnap points out [Carnap, 1950], some of the controversy concerning the support of empirical hypotheses by data is a result of the conflation of two distinct notions. One is the total support given a hypothesis by a body of evidence. Carnap's initial measure for this is his c*; this is intended as an explication of one sense of the ordinary language word “probability.” This is the sense involved when we say, “Relative to the evidence we have, the probability is high that rabies is caused by a virus.” The other notion is that of “support” in the active sense, in which we say that a certain piece of evidence supports a hypothesis, as in “The detectable presence of antibodies supports the viral hypothesis.” This does not mean that that single piece of evidence makes the hypothesis “highly probable” (much less “acceptable”), but that it makes the hypothesis more probable than it was. Thus, the presence of water on Mars supports the hypothesis that that there was once life on Mars, but it does not make that hypothesis highly probable, or even more probable than not.

Whereas c*(h, e) is (for Carnap, in 1950) the correct measure of the degree of support of the hypothesis h by the evidence e, the increase of the support of h due to e given background knowledge b is the amount by which e increases the probability of h: c*(h, b Λ e) – c*(h, b). We would say that e supports h relative to background b if this quantity is positive, and undermines h relative to b if this quantity is negative.

In this chapter we briefly discuss the goals of cryptography (Section 1.1). In particular, we discuss the basic problems of secure encryption, digital signatures, and fault-tolerant protocols. These problems lead to the notions of pseudorandom generators and zero-knowledge proofs, which are discussed as well.

Our approach to cryptography is based on computational complexity. Hence, this introductory chapter also contains a section presenting the computational models used throughout the book (Section 1.3). Likewise, this chapter contains a section presenting some elementary background from probability theory that is used extensively in the book (Section 1.2).

Finally, we motivate the rigorous approach employed throughout this book and discuss some of its aspects (Section 1.4).

Teaching Tip. Parts of Section 1.4 may be more suitable for the last lecture (i.e., as part of the concluding remarks) than for the first one (i.e., as part of the introductory remarks). This refers specifically to Sections 1.4.2 and 1.4.3.

Cryptography: Main Topics

Historically, the term “cryptography” has been associated with the problem of designing and analyzing encryption schemes (i.e., schemes that provide secret communication over insecure communication media). However, since the 1970s, problems such as constructing unforgeable digital signatures and designing fault-tolerant protocols have also been considered as falling within the domain of cryptography. In fact, cryptography can be viewed as concerned with the design of any system that needs to withstand malicious attempts to abuse it.