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Berkeley STAT 157 - Mathematical Analysis of Subjectively Defined Coincidences

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Mathematical Analysis of Subjectively DefinedCoincidences; a case study using WikipediaDavid J. Aldous∗Department of Statistics367 Evans Hall # 3860U.C. Berkeley CA [email protected]/users/aldouset al.September 23, 2008AbstractRationalists assert that real-life coincidences occur no more fre-quently than is predictable by chance, but (outside stylized settingssuch as birthdays) empirical evidence is scant. We describe a study,with a few real-life features, of coincidences noticed in reading ran-dom articles in Wikipedia. Part of a rationalist program (that one canuse specific observed coincidences to infer general types of unobservedcoincidence and estimate probabilities of coincidences therein) can beexamined in this context, and fits our data well enough. Though thisconclusion may be unremarkable, the study may provide guidance forthe design of more “real-life” studies of coincidences.xxx in progress - started with Fayd Shelley in 2006, continuingwith Sunny Zhao in 2008∗Research supported by N.S.F Grant DMS-020306211 IntroductionA long and continuing tradition outside mainstream science [1, 3, 5] as-signs spiritual or paranormal significance to coincidences, by relating storiesand implicitly or explicitly asserting that the observed coincidences are im-mensely too unlikely to be explicable as “just chance”. Self-described ratio-nalists dispute this, firstly by pointing out that (as illustrated by the wellknown birthday paradox [7]) untrained intuition about probabilities of coin-cidences is unreliable, and secondly by asserting that (in everyday language)observing events with a priori chances of one in a gazillion is not surprisingbecause there are a gazillion possible other such events which might haveoccurred. While the authors (and most readers, we imagine) take the ratio-nalist view, it must be admitted that we know of no particularly convincingstudies giving evidence that interesting real-life coincidences occur no morefrequently than is predictable by chance. The birthday paradox analysis isan instance of what we’ll call a small universe model, consisting of an ex-plicit probability model expressible in abstract terms (i.e. the fact that the365 categories are concretely “days of the year” is not used) and in which weprespecify what will be counted as a coincidence. Certainly mathematicalprobabilists can invent and analyze more elaborate small universe models,but these miss what we regard as three essential features of real-life coinci-dences:(i) coincidences are judged subjectively – different people will make differentjudgements;(ii) if there really are gazillions of possible coincidences, then we’re not goingto be able to specify them all in advance; – we just recognize them as theyhappen;(iii) what constitutes a coincidence between two events depends very muchon the concrete nature of the events.In this paper we seek to take one tiny step away from small universe modelsby studying a setting with these three features.Almost the only serious discussion of the big picture of coincidences froma statistical viewpoint is Diaconis-Mosteller [2]. Our “gazillions” explana-tion, which they call the law of truly large numbers and which is also calledLittlewood’s law [9], is one of four principles they invoke to explain coinci-dences (the others being hidden cause; memory, perception or other psy-chological effects; and counting close events as if they were identical). Theysummarize earlier data in several contexts such as ESP and psychology ex-periments, mention the extensive list of coincidences recorded by Kammerer[4], show a few “small universe” calculations, and end with the conclusion2In brief, we argue (perhaps along with Jung) that coincidencesoccur in the mind of observers. To some extent we are handi-capped by lack of empirical work. We do not have a notion of howmany coincidences occur per unit of time or how this rate mightchange with training or heightened awareness. . . . Although Jungand we are heavily invested in coincidences as a subjective mat-ter, we can imagine some objective definitions of coincidencesand the possibility of empirical research to find out how fre-quently they occur. Such information might help us.Let’s take a paragraph to speculate what a mathematical theory of real-life coincidences might look like, by analogy with familiar random walk/Brownianmotion models of the stock market. Daily fluctuations of the S&P500 indexhave a s.d. (standard deviation) of a little less than 1%. Nobody has anexplanation, in terms of more fundamental quantities, of why this s.d. is 1%instead of 3% or 0.3% (unlike physical Brownian motion, where diffusivityrate of a macroscopic particle can be predicted from physical laws and theother parameters of the system). But taking daily s.d. as an empirically-observed parameter, the random walk model makes testable predictions ofother aspects of the market (fluctuations over different time scales; optionprices). By analogy, the observed rate of subjectively-judged coincidences insome aspect of real life may not be practically predictable in terms of morefundamental quantities, but one could still hope to develop a self-consistenttheory which gives testable predictions of varying aspects of coincidences.The aspect we study is single-affinity coincidences, exemplified in reallife by stories such asIn talking with a stranger on a plane trip, you discover you bothattended the same elementary school, which is in a city not onthat plane route.Call this (“same elementary school”) a specific coincidence; one might plau-sibly estimate, within a factor of 2 or so, the a priori probability of sucha specific coincidence. Now a specific coincidence like this suggests a coin-cidence type, in this case “having an affinity (both members of some rela-tively small set of people) with the stranger”, where the number of possibleaffinities (attended first ever Star Trek convention; grow orchids; mothersnamed Chloe) is clearly very large and subjective. Nevertheless one couldtry to estimate (within a factor of 10, say) the chance of some coincidencewithin this coincidence type. Next one can think of many different specificsingle-affinity coincidences (finding a dollar bill in the street, twice in one3day; seeing on television someone you know personally) which should beassigned to different types, and it is hard to imagine being able to writedown a comprehensive list of


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