Chapter 8The Local UniformityPrincipleAs a change of style, this lecture concerns some conceptual/foundationalaspects of probability which can be illustrated with real examples. As willbecome clear in Lecture xxx, traditional discussions of “philosophical foun-dations of probability and statistics” emphasize topics that do not interestme while ignoring topics that do, and I am using this chapter (only) toexplain a few of the latter.I use “throwing a dart at a target” as a paradigm example of one kindof “physical” randomness. Figure 1 shows th e result of 100 throws1at adartboard – the kind of experiment you could easily repeat yourse l f. Toshow scale, we draw an imaginary playing card centered at the center ofthe board. Textbooks are apt (cf. section 1.4) to give examples of the kind“suppose the throws land uniformly (or as bivariate Normal)” but neithersupposition is remotely accurate.8.1 Four comments on dart throwsThe followin g comments are trite in the context of darts, but are (to me)interesting starting points for discussion withi n other contexts, as I will tryto show in the rest of the lecture.1By a student Beau La Mont, who was aiming at the center of the board. One missedthe board.7980 CHAPTER 8. THE LOCAL UNIFORMITY PRINCIPLEFigure 1. 99 dart throws, centered on a 2.25” × 3.5” playing card.Darts are not dice. As the next p oi nt says, the result of dart throws area combination of skill and chance, rather than the “pure chance” of a dieroll, and so the chances vary between individual throwers, unlike dice. Andthe “physical symmetry” of a die or analogous gaming artifact allows one toassert that different outcomes are equally likely, which has no counterpartfor dar t throws.Darts provide a vivid and quantifiable ins tan ce of the luck-skillcombination. Whether one can quantify the relative contributions of skilland luck to observed success, in some particular field of human endeavor orfor some individual person, is one of the mos t intrigu i ng aspects of proba-bility in the real world2. For darts one can quantify skill as, for inst anc e,2But I haven’t managed to write a satisfactory lecture on this topic.8.1. FOUR COMMENTS ON DART THROWS 81mean-square deviation from target point, and presumably there is strongcorrelation b e tween that statis t i c and success at traditional games based ondart throws. In mnay professional team sports there are statistics for in-dividual player (e.g. pitcher or quarterback) per form anc e , which again arepresumably correlated with the player’s contribution to team success. Butmoving on to entrepreneurs or movie stars, it is hard to know what “skill”statistic can be measured, to compare with some quantitative measure ofsuccess.It seems perfectly reasonable to model dart throws via densi tyfunctions. A dart throw is a textbook example of a random point in twodimensions whose distribution is assumed to be described by some den si tyfunction. What this means, in words, is th at for any two points z and z�thatare sufficiently close, the chance of the dart landing very close to z is almostthe same as the chanc e of landing very close to z�. This seems so reasonablethat it is rarely commented upon. A main point of t hi s lecture is that in factthis kind of assumption, which is made in many contexts and which I willcall the local uniformity principle, has more conse q ue nc es than one mightimagine. Let me contrast with contexts where one assumes indepen dence,inwhich case we are fully aware that independ en ce is an assumption (maybejustifiable on intuitive or empirical grounds) and that the reliability of anymathematical consequence we might derive is linked to the reliability of theassumption.It seems perfectly reasonable to model dart throws as IID. Onecan think of several reasons t h e IID model might not be accurate.Thefirstfew throws may involve learning adjustments, and eventually the throwermight get tired or bored. But aside from such specific effects, the IID modelseems conceptually reasonable. Now freshman textbooks seem to leave theimpression (amongst non-professional statisticians) that any list of data3canbe regarded as IID samples from some probabili ty distribution – if you doa textb ook test of significance or confidence interval, then you are model i ngthem as IID. But consider for instance the areas of the 50 U.S. States, or theunemployment rates in each State next year, or the change in unemploymentrate between this year and next year. To me there i s no reason to think theseare independent (in the sense of probability theory) between States. I willreturn to th is issue in section 8.5.3Implicitly, the sa m e attribute of different “individuals”.82 CHAPTER 8. THE LOCAL UNIFORMITY PRINCIPLE8.2 County fairThis section treats a con cr et e setting which will seem very special, but in thenext two sections I will show that the abstract idea applies more broadly.There is a fairground game in which playing cards4are stuck to a largeboard in a regular pattern, with space between cards. See Figure 2. Youpay your dollar, get three darts, and if you can throw t h e darts and makethem stick into three different cards then you win a small prize.0 1 2 3 4 5 6 inchesFigure 2. Playing cards on a wall. The playing cards on the left are “bridgesize” 2.25 by 3.5 inches, wit h spacing 1 inch between rows. The wall is much l ar gerthan shown, with hundreds of cards attached. In the center is the “basic unit” ofthe repeatin g pattern. On the right is the pattern shrunk by a fac t or of 3.One could conduct time-consuming experiments with throwers of differ-ent skills and different patterns. But – for the point I want to make – Ican be lazier and work with the previous data set of 100 throws by Beau,and see what would have happened with differently scaled cards. 36 throwswould have hit a normal sized card as target, so we estimate the proba-bility as 0.36. The • in F i gur e 3 show how this probability increas es withthe card size. As one expects, this probability is near zero for a postagestamp size and near one for a p aper back size. If we repeated the experimentwith a different