Ten Great Ideas About Chance a Review by Mark Huber
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BOOK REVIEW Ten Great Ideas about Chance A Review by Mark Huber Ten Great Ideas About Chance Having had Diaconis as my professor and postdoctoral By Persi Diaconis and Brian Skyrms advisor some two decades ago, I found the cadences and style of much of the text familiar. Throughout, the book Most people are familiar with the is written in an engaging and readable way. History and basic rules and formulas of prob- philosophy are woven together throughout the chapters, ability. For instance, the chance which, as the title implies, are organized thematically rather that event A occurs plus the chance than chronologically. that A does not occur must add to The story begins with the first great idea: Chance can be 1. But the question of why these measured. The word probability itself derives from the Latin rules exist and what exactly prob- probabilis, used by Cicero to denote that “...which for the abilities are, well, that is a question most part usually comes to pass” (De inventione, I.29.46, Princeton University Press, 2018 Press, Princeton University ISBN: 9780691174167 272 pages, Hardcover, often left unaddressed in prob- [2]). Even today, modern courtrooms in the United States ability courses ranging from the shy away from assigning numerical values to probabilities, elementary to graduate level. preferring statements such as “preponderance of the evi- Persi Diaconis and Brian Skyrms have taken up these dence” or “beyond a reasonable doubt.” Those dealing with questions in Ten Great Ideas About Chance, a whirlwind chance and the unknown are reluctant to assign an actual tour through the history and philosophy of probability. number to the chance that an event occurs. Diaconis is the Mary V. Sunseri Professor of Statistics and The idea that chance could be measured quantitatively Mathematics at Stanford University, and has long been by a number took until the sixteenth century to arise. For known for his seminal work on randomization techniques instance, consider the problem of finding the probability such as card shuffling. Skyrms is a professor of philosophy at Stanford and Distinguished Professor of Logic and Phi- losophy of Science and Economics at the University of Cal- ifornia at Irvine. His recent work has been in evolutionary game theory. The book they created is based upon a course they have taught at Stanford for the past ten years, and is aimed at a reader who has already gone through a typical undergraduate course in elementary probability. Mark Huber is the Fletcher Jones Foundation Professor of Mathematics and Statistics at Claremont McKenna College. His email address is mhuber @cmc.edu. Communicated by Notices Book Review Editor Stephan Ramon Garcia. For permission to reprint this article, please contact: reprint-permission @ams.org. DOI: http://dx.doi.org/10.1090/noti1888 Figure 1. A six-sided die. JUNE/JULY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 917 Book Review of rolling a 1 or a 2 on a fair probabilities, or what prob- six-sided die (see Figure 1). abilities actually are. They This classic problem is usually treat probability as a measure, solved by assuming that each but they leave unanswered outcome of the die is equally the question of what exactly likely. Therefore, since there probability measures. are six sides and probability Now both of the authors must sum to one, each has are unabashed Bayesians, and a one in six chance of occur- upfront about this through- ring, and the probability of out the work. It is therefore a 1 or 2 is two out of six, or unsurprising that their an- one third. swers to this question lie with Even in the Middle Ages, Bayesian philosophy. In the though, mathematicians such Bayesian approach to chance, as Cardano were knowledge- what probabilities measure able about the cheating ways is the degree of belief by an of gamblers, and considered individual that the event will what might happen with occur. Given an initial set of shaved or otherwise altered beliefs, one can update their dice. Gambling drove prog- beliefs as new information ress in probability: A histor- about the events comes in. ical tidbit offered up in this This is the center of the Bayes- chapter concerns Galileo ian philosophy. being asked by his patron to The first Bayesian idea, and analyze a particular gambling the second great idea about problem to discover who had Figure 2. Blaise Pascal. chance in the book, is judg- the advantage. ment, the idea that probabili- A leap forward in understanding how to measure ties arise from individual beliefs about how likely an event probability came about during a correspondence between is to occur. But how can we ascertain what an individual’s Pascal (Figure 2) and Fermat in the 1600s concerning the belief really is? And how do we derive the logical rules of problem of points [4]. In this problem, two equally skilled probability from this point of view? opponents begin a series of games to decide who obtains a One way is to use an idea in mathematical finance. Sup- certain amount of prize money. Each winner is assigned a pose I have a security that pays $1.00 if event A occurs, and point, and the first to reach a fixed number (set in advance) nothing if A does not occur. We use AC to mean the event of points wins the entire prize. Suppose they play the first that A does not occur, also game and the first player wins. At this point their match is called the complement of A. interrupted, and so the question is: How should they split How much should I be willing the prize money given this information? to pay to purchase this security? The discussion of Pascal and Fermat is the first time the Well, the expected payoff from concept of expected value appears. The expected value is owning this security is just the the average amount of money that each player would walk probability that event A occurs. away with in their situation, and is how they should split If I judge A to have probability the prize to achieve a fair division. Use of expected value 0.47, then I should be willing Figure 3. A simple security. enables the consistent development of probabilities where to pay $0.47 to own one share the outcomes are not all equally likely. of the security (see Figure 3). This was the beginning of turning chance into something Now suppose I do the same thing for several events. A numerical, something that could be dealt with mathemati- Dutch book is a portfolio of multiple securities that guaran- cally rather than as a vague notion of plausibility. This path tees that the owner gains money. A book is the list of odds reaches its culmination in the fifth great idea: the formal that a casino or racetrack would post before races. The definition of a probability measure within a set-theoretic person making the book is still known as a bookmaker or framework. The definition introduced by Kolmogorov is bookie. The origins of the Dutch part of the term are lost now a standard feature of introductory texts. to history. These definitions allow for proving theorems and pro- As an example of a Dutch book, suppose the security vide a set-theoretic foundation of the field, but they give for A costs $0.30 and the security for AC costs $0.60. With little insight into why this is the proper way to calculate these prices I could buy one security that pays $1.00 when A 918 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 6 Book Review occurs and nothing when AC occurs, plus I can buy a second also the probabilities. This updated probability is called security that pays $1.00 when AC occurs and nothing when the posterior. A occurs. Together, these two securities form a Dutch book If I then ran a second experiment, the posterior from that costs $0.30 + $0.60 = $0.90. If A occurs, the first secu- the first experiment would become the prior for the second rity returns $1.00 and the second security returns nothing. experiment, and so on. In this way, I could keep multiplying If AC occurs, then the second security returns $1.00 and Bayes’ Factors and updating my beliefs about the veracity the first security returns nothing. Either way, the Dutch of theory X versus theory Y for as long as I keep getting book makes $1.00, but at a cost of only $0.90. Therefore I more evidence. am guaranteed to receive $0.10 regardless of the outcome. What about the practitioners who do not hold that Bruno de Finetti [3] called probabilities associated with probabilities are degrees of belief? Is it a coincidence that such a Dutch book incoherent because prices (and proba- they use the same rules and definitions of probability? The bility beliefs) would naturally change if this type of profit seventh great idea provides one answer to this question. taking were possible. Either the price of A or AC would rise This idea is Bruno de Finetti’s concept of exchangeability. until the possibility of a Dutch book disappeared. This kind Exchangeability is a type of symmetry applied to se- of argument leads directly to the notion that the probability quences of random variables. For instance, consider a coin C of A and A should add to 1. that lands heads or tails, H or T, on each flip. Then applying The idea here is not to derive the commonly used defini- Bayes’ Theorem, no matter what your prior is, if the coin tions of a probability measure, but instead to explain why is flipped three times, the sequences HHT, HTH, and THH probabilities behave in the way that they do.