How and Why Did Probability Theory Come About?
HHow and Why Did Probability Theory ComeAbout? by Nozer D. SingpurwallaThe City University of Hong Kong, Hong Kong,and The George Washington University, Washington, D.C.Boya LaiThe City University of Hong Kong, Hong KongMarch 9, 2020
Abstract
This paper is a top down historical perspective on the severalphases in the development of probability from its prehistoric originsto its modern day evolution, as one of the key methodologies in ar-tificial intelligence, data science, and machine learning.It is written in honor of Barry Arnold’s birthday for his many contri-butions to statistical theory and methodology. Despite the fact thatmuch of Barry’s work is technical, a descriptive document to markhis achievements should not be viewed as being out of line. Barry’sdissertation adviser at Stanford (he received a Ph.D. in Statisticsthere) was a philosopher of Science who dug deep in the foundationsand roots of probability, and it is this breadth of perspective is whatBarry has inherent. The paper is based on lecture materials com-piled by the first author from various published sources, and overa long period of time. The material below gives a limited list ofreferences, because the cast of characters is many, and their contri-butions are a part of the historical heritage of those of us who areinterested in probability, statistics, and the many topics they havespawned. 1 a r X i v : . [ s t a t . O T ] M a r Overview
The material here attempts to give a top down historical perspective on theseveral phases in the evolution of probability, from its prehistoric origins forimperial needs, to its current state as a branch of mathematics. As a branchof the mathematical sciences, probability evolved in five stages, not countinga period of stagnation between the second and the third stages, when doubtswere cast about its relevance as a mathematical discipline. Also pointed outare paradoxes in probability, spawned by the absence of its precise definition,leading to its last two phases, namely, an axiomatic and a subjective view ofprobability.
The rulers of ancient Egypt, Greece, and Rome, collected census data for taxes,grain distribution, and other matters of administration; this activity certaintyhad an impact on the origins of probability.Next to come was the “Doomsday” list of William the Norman (1027 –1087), which was so exhaustive an economic survey that it reminded one ofthe final and the last judgment by God in the Christian faith. Following thiswere the “London Bills of Mortality”, published since 1517, and notions such asthe chance of death in a given time period, the chance of survival to a certainage, and the like, originated, around about 1535, almost a century before JohnGraunt’s celebrated mortality table. Another impetus to the origins of probabil-ity came from marine insurance in the 1300’s, and also during the renaissance,wherein an emphasis was placed on observation and experiments in the naturalsciences – especially, on errors of observation.From a philosophical angle, the interrelations between chance and causalityhave been on the philosopher’s agenda since the ancient times. These too hadan impact on the origins of probability. In 1292, a treatise on the theory of thelogical ideas of
Syadvada (which is the basis of India’s Jaina religion), lists sevenpredications of which the fourth supplies a foundation for modern probability.Another influential angle was the famous dictum of Thomas Hobbs (1588 –1679), whose thesis was that no matter for how long we observe a phenomenon,this is not sufficient grounds for its absolute and definitive knowledge.To summarize, the prehistoric impact on probability came from: census,commerce, renaissance, scientific observation, and philosophy.
Apart from the discussion above, there is another belief that probability theoryowes its birth to gambling. To some, this is a questionable issue. They claimthat since gambling has been practiced since 5000 BC, it could not have taken6000 years for it to influence probability. Their view is that it was commercethat really influenced the development of probability.2onetheless, gambling has had an impact on probability, and its earliesttraces are in the literature, such as “
De Vetula ” of Richard de Fournival (1200-1250), and Dante’s “
Divine Comedy ” (1307 – 1321), wherein combinatorialarguments pertaining to outcomes of games of chance were mentioned.Paccioli (1445 -1514) published in 1487 “
Summa de Arithmetica, Geometria,Proportioni et Proportionalita ”, which was an encyclopedia of the mathematicalknowledge of his period in Venice, and in the section labelled “unusual prob-lems”, he discussed the question of the fair division of stakes when a match isstopped in advance of an agreed termination of the game. Paccioli’s solutionembeds notions of probability. This is also called the “ problem of points ”, andwas a trigger point of the famous Pascal Fermat correspondence.Cardano (1501 – 1576), and Tartagalia (1499 – 1557) contributed much to theconnection between probability and gambling. Cardano developed probabilisticnotions in “The Book on Games of Chance”, written in 1526, as “
Liber deLudo Aleae ”. In this book, Cardano enumerates possibilities, permutations,deviations of frequencies from “portion”, introduces the notions of fair gamesand expectation, equally likely events, and uses the addition and multiplicationrules of probability for independent events. He even came close to inventing thelaw of large numbers. However, Cardano was an ardent gambler who restrictedhis writings, only to games of chance. All the same, as one can surmise, he setthe stage for much that was to follow.Tartagalia (1499-1557) published in Venice in 1556, his treatise on “Numberand Measure” in which he related problems of probability to those of combi-natorics, and offered correct solutions to the problems posed by Paccioli, inparticular, the problem of the division of stakes, (or the problem of points).Following Cardano and Tartagalia, was Galileo (1564 – 1642), who posited thaterrors of measurement are inevitable; they are symmetric, and clustered arounda true value. He in fact revealed many of the characteristics of the normalprobability distribution.The above developments, perhaps mark the end of the phase of the earliestwritings on probability, subsequent to its prehistoric phase.
This phase can be categorized into five stages, and includes a phase called “theperiod of stagnation”, between the second and the third stage, when concernswere raised about probability as a branch of mathematics. Also included is aphase labeled “paradoxes in probability”, which can be seen as the doorstepto the development of the last two stages in the evolution of probability as amathematical discipline.Within the five stages alluded to above, are also some milestones in theevolution of statistics, which evolved as a way to reason with numbers.3 .1 Stage I. Of the Development of Probability as a Sci-ence
Up until the middle of the 17th century, there were no general methods forsolving probabilistic problems. Specific problems had been solved, and a sub-stantial amount of knowledge was accumulated. The term probability (nor itsdisposition as a number) was not a part of the lexicon in the solution of suchproblems.In the middle of the 17th century some prominent mathematicians like Pas-cal, Fermat, and Huygens became involved in the development of probability,even without mentioning the term. These individuals were familiar with Car-dano’s addition and the multiplication rules, the notion of independence, andput to practice the notion of expectation using combinatorics. They developednew methods for solving problems, determined the realm of problems to whichthis new science is applicable, and in so doing were on the verge of transformingprobability to a bona fide science.Two very important and key individuals need to be mentioned in Stage I.They were: Chevalier de M´ere, and Christian Huygens. They brought prob-ability into a new stage as a science. Chevalier de M´ere (1607 – 1684) wasa philosopher and a man of letters; he wrote to Pascal about the division ofstakes [considered by Paccioli but per Shafer (2019) solved in the 1400’s by twoItalian abacus masters, and his (de M´ere’s) solution to it. With Pascal andFermat, he had authored in 1662 “
Ars Cognitandi ” (Art of Thinking) as a partof the Arnold - Nicole (who were abbots at the Port Royal Monastry), a bookon “
Port RoyaleLogic ”. de M´ere’s letter to Pascal triggered a correspondencebetween Pascal and Fermat in 1654, and thus originated the founding documenton mathematical probability.Even though many mathematicians of that period devoted much attentionto the solution of games of chance, actual gambling was condemned. Thus, themyth that Chevalier de Mere was a fervent gambler. Rather, he was a manof letters who viewed probability only as a “useless curiosity”. By contrast,Cardano, who was an ardent gambler, used mathematics for gambling, but in1526 did not quite hit upon the notion of probability as a number.Christian Huygens (1629 – 1695), a Dutchman from Holland, visited Parisin 1655 to receive a doctorate in law. He was impressed with the problems ongambling of Pascal and Fermat, and undertook further work on it. He was told ofthe solutions but not the methods (which were published posthumously, becauseboth Pascal and Fermat posed problems to each other but hid their methods ofsolution). The correspondence between Pascal and Fermat was published onlyin 1679.Huygens returned to Holland and begun work on solving the problems posedby Pascal and Fermat). Huygens solutions, independent of the methods ofPascal and Fermat, but identical to those of Pascal and Fermat, were publishedin his book (written in Latin) called “
About Dice Games ”. This book appearedin 1657 wherein Huygens says “... we are dealing not only with games but ratherwith a foundation of a new theory, both deep and interesting.” His reasons for4riting this book was the absence of methods used by Pascal and Fermat.This book is viewed as the first published treatise on mathematical proba-bility. Huygens book can be viewed as being the first format document on theintroduction of mathematical probability, until Bernoulli’s famous “
Ars Con-jectandi ” (a possible imitation of the Pascal - Fermat- de Mere’s, Ars Cogni-tandi). Huygen’s book was also the first to introduce and to apply the notion ofexpectation in commercial and industrial problems. Huygen’s terminology wascommercial.Subsequent to the above, more and more works on probability began toappear,most notable being the birth to a new discipline, now called “
Data Sci-ence ”. In 1662, at about the same time as Huygen’s book, John Graunt, anEnglishman, published a tiny book devoted to problems of vital statistics. Huy-gens was asked to comment on this landmark book, which he did favorably.Indeed, in 1669, using Graunt’s work Huygens constructed a mortality curve,and initiated the application of probability to demography, and to annuities. In1690, another Englishman, by the name of William Petty published his treatieson “
Political Arithmetic ”, which was about a method of reasoning on mattersof government, via the use of numbers. This can now be seen as a foundingdocument on Government Statistics.Preceding Petty’s treatise, was work on actuarial mathematics and the worthof annuities, due to deWitts in 1671, followed by that of Edmund Haley in1693, who published the very first mortality table based on data from Breslau.Between (1791 - 1799), a Scotsman named John Sinclair published 21 volumesof his Statistical Account of Scotland, and introduced the word “
Statistics ” toreplace Petty’s political arithmetic. Up until 1796 the word “statistics” was usedin Germany to describe the political strength, happiness, and the improvementof a country, as a measure of its well-being. Statistics was an artificial word,with no evidential meaning, that is now used for anything having to do withdata. Sinclair used it to garner attention over Petty’s political arithmetic, whichdid not seemed to have gained traction [cf. von Collani (2014).To summarize, Huygens recognized the role of probability as a science, wrotethe first book on it, applied the notion of expectation to commerce and industry,and used probability for assessing demography and insurance. Huygens’ Bookplayed an important role in the history of probability. Jacob Bernoulli, who in-troduced the term “probability”, based on the Latin “ probabilitas ”, was greatlyinfluenced by Huygen’s book. Bernoulli’s work established the foundations ofmathematical probability.Bernoulli’s word, probability is based on the term “ probabilitas ”, which wasa moral system of the Catholic Church. Probabilitas was formally introducedin 1577 by the Spanish Dominican, Bartholome de Medina, and was mainlyapplied by Jesuit priests. Bernoulli’s aim in writing Ars Conjectandi was tointroduce a new branch of science, that he called Stochastics, or the science ofprediction.To Bernoulli, a relevant feature of “stochastics” was an event’s readinessto occur , and ”probability” , the degree of certainty of its occurrence, seevon Colani (2014). Thus, to Bernoulli, stochastics was the art of measuring5robability as exactly as is possible.However, Bernoulli acknowledged that the determination of the true value ofprobability is impossible, and labeled as “mad” any attempt at doing so. Thismotivated him to develop his law of large numbers, as an empirical method todetermine a lower and an upper limit for an unknown probability. Note thatBernoulli’s notion of probability was devoid of any mathematical basis.Given below is a graphic of the evolution of mathematical probability, upuntil the beginning of Stage II which established it as a mathematical science.Figure 1: Evolution of Mathematical Probability James (Jacques) Bernoulli (1654 -1705) in his 1713 “Ars Conjectandi” provedthe first limit theorem, and in so doing, raised the status of probability to thatof a formal mathematical science. This book was published posthumously byhis nephew Nichols (1) Bernoulli [who also applied probability to matters ofjurisprudence, like the credibility of a witness].The contribution of Bernoulli to make probability a bona fide mathematicalscience is that he interpreted propositions in Huygen’s book, showed inapplica-bility of the addition law to non-disjoint events, gave the binomial formula, and6sed Leibnitz’s combinatorics for solving probability problems.He proved the weak law of large numbers as a way to bound a “true” prob-ability, and interpreted probability as the degree of certainty of an event’s oc-currence. Bernoulli was a metaphysical determinist; i.e. if we know the positionof a dice, its speed, its distance from the board, etc. we can exactly predictits outcome. Thus, to Bernoulli, probability, or chance, the terms he used in-terchangeably, depends on our state of knowledge, and is thus personal to theindividual specifying it.However, since all knowledge is not possible, we assume a statistical regu-larity in a large number of trials, say n , and conclude that for the tossing ofcoins, the deviation of m/n from p, as n → ∞ , is small with a large probability; m is the total number of heads in the n trials. Bernoulli also touched uponthe philosophical problems connected with probability, and asserted probabilityshould also be applied to situations outside games of chance.Besides Bernoulli there were others who worked on probability during thebeginning of the 18-th century. We name a few.Pierre de Montmort (1678 – 1719), a mathematician, who was chosen byLeibniz on the commission to inquire about the priority of inventing differentialand integral calculus between him and Newton; de Montmort favored Newton;see Maistrov (1974). His basic work on probability entitled “ Essai d’Analysesur les Jeux de Hazard ” was published in 1708 (5 years prior to Bernoulli’sposthumously published work). It was in a letter to de Montmort that Nicholas(1) Bernoulli posed the St. Petersburg paradox. Montmort’s main effort wasthe applications of probability to human behavior.Abraham de Moivre’s (1667 – 1754) principal work in probability was in “
TheDoctrine of Chances ”, 1718. Here, without addressing the matter of what prob-ability is, de Moiver discusses topics connected to Bernoulli’s theorem, and theproblem of the duration of the play (first proposed by Huygens). de Moivre in-vestigated the probabilities of various deviations between m/n and p, for p=1/2.Laplace extended these to p ∈ (0,1), and thus the de Moivre-Laplace theoremis the second limit theorem in mathematical probability.Thomas Bayes (1792 – 1761), speculated as being tutored by de Moivre, pub-lished his famous essay posthumously in 1763; it was entitled “ Thomas Bayes ’ sEssay Towards Solving a Problem in the Doctrine of Chances ”; it addressedthe following question: what is the chance that p ∈ (a,b) given x and n? Bayesoffered a solution to this problem using solely the calculus of probability. Inso doing, he introduced the notion of what is referred to as “ probabilisticinduction ”.To obtain his solution, Bayes used what is now called Bayes formula (whichis really an alternative form of the well-known, by then, multiplication rule),interpreted conditional probability and its subtleties, and assumed a uniformdistribution on p (via eliciting priors on the observables – i.e. the predictivedistribution). It was Laplace who coined the term “Bayes Theorem” and set innotion this terminology – Bayes did not invent Bayes Theorem.Daniel Bernoulli (1700 – 1782) introduced the idea of probability curves,and applied differential calculus to problems of probability theory, and in so7oing simplified many of the cumbersome combinatoric formulas used before.However, his most important contribution is the introduction of the notion of“ utility ” or “ moral expectation”, and its use in solving the St. Petersburgparadox, posed by Nicholas (1) Bernoulli.Condorcet [Jean Antoine de Caritat, Marquis de Condorcet] (1743 – 1794)was a well-known sociologist and economist during the period of the FrenchRevolution. His main contribution is his introduction of the notion of “ proba-bilite’ propre ”, which is a subjective, or personal, probability. His ideas wererejected as being beyond the scope of mathematical probability theory.After Bernoulli, one of the great minds that came to wrestle with probabilitywas Pierre Simon. de Laplace. His main technical contribution is the de Moivre-Laplace central limit theorem, for Bernoulli trials. His contribution to largerissues is extending the realm of applicability of probability to social phenomena,and his reinforcement of Condorcets notion of subjective probability. That is“probability is relative in part to ignorance, and our knowledge”. If a coin isasymmetrical, but we do not know which side, then its probability of head is1/2. Laplace also played a role in developing statistics.Stage II of the development of probability ends with Gauss (1777 – 1855),who derived the normal law for the distribution of errors. [This was also doneby Robert Adrian (1755 – 1843) an obscure American mathematician].Poisson (1781 -1840) did much work on technical and practical aspectsof probability. He subscribed to the subjectivie view of probability, and likeLaplace felt that probability can also be applied to jurisprudence. Poisson’smain contributions is a generalization of Bernoulli’s theorem when the proba-bility of an event changes from trial to trial, so that if ˜ p is the arithmetic meanof these probabilities, then lim n →∞ P ( | mn − ˜ p | < (cid:15) ) = 1,and his proof that as p n → n → ∞ , P ( m/n ) = e − n m ! e − λ , where λ = np n , the famous Poisson formula; recall that m is the number of events inn Bernoulli trials. The period (1860-1900) is also viewed as one of stagnation in the development ofprobability. Many felt that its application to social problems was a compromisein the mathematical sciences.The areas of application being not clearly defined there was much controversyabout the subject. There was much criticism of the early developers, like Pascal,Bernoulli, Laplace, and Poisson for their subjectivist inklings via metaphysicaldeterminism. The period of stnagation terminated with the emergence of thenow famous Russian School of probability.8 .4 Stage III. Creation of The Russian School
The originators of the Russian School in probability were Ostrogradski (1801-1862), and Bunyakowski (1804-1889). Ostrogradski, influenced by Laplace, wasa proponent of the principle of insufficient reason, and applied the theory ofprobability to moral problems. He too subscribed to the notion that probabilityis a measure of our ignorance, and is thus subjective. Bunyakowski wrote thefirst Russian book in probability, and introduced the needed terminology; hetoo was a determinist in the spirit of Bernoulli and Laplace.Chebyshev (1821-1894), influenced by Ostrogradski and Bunyakowski, iscredited with the creation of the Russian school of probability. His studentsMarkov, Voroni, Lyapunov, and Steklov, pushed frontiers of the subject to themodern era. In effect, Chebyshev and his followers, broke the period of stagna-tion and impasse in probability, as a mathematical science. Chebyshev definedthe subject matter of probability theory as the mathematical science of con-structing probabilities of an event based on probabilities of other events. He didnot discuss how initial probabilities are to be obtained. Chebyshev introducedmathematical rigor in the theorems, and obtained exact estimates or inequali-ties of derivations from limiting laws which arise when the number of trials islarge but finite.Philosophically, Chebychev and his followers were materialists through thenatural sciences, mechanics and, mathematics. They were guided by the opinionthat only those investigations initiated by applications are of value, and onlytheories which arise from a consideration of particular cases are useful. Thematerialist philosophy is founded on the belief that nothing exists but matteritself and its manifestations.Markov (1856-1922) was Chebychev’s closest disciples and his most colorfulspokesperson. He transformed probability, with clarity and rigor, to one ofthe most perfect field in mathematics. His noteworthy works are on the limittheorems for sums of independent and dependent random variables using themethod of moments. Markov introduced the famous chain named after him, foranalyzing Pushkin’s poem “
Eugene Onegnin ”.Lyapanov (1857-1918) improvised on the proofs of Markov’s theorems usingcharacteristic functions; the central limit theorem is named after him. Lindbergand Feller later improved on Lyapunov’s theorems.
The evolution of probability as a mathematical science would not completewithout a mention of its impact in physics, one of the most basic of all thesciences. In 1827, Robert Brown, an English botanist, detected the movementof minute suspended particles in an unpredictable manner. This movement isdue to random bombardments of chaotically moving molecules in suspension.Using probabilistic arguments, Albert Einstein in 1905 was able to develop asound theory for such motions. It was observed that every sufficiently smallgrain suspended in a fluid constantly moves in an unpredictable manner.9f before the second half of the 19-th century, the basic areas of applicationof probability were in the processing of observations, the second half was inphysics. This was prompted by the work of Ludwig Boltzman (1844-1906), anAustrian, and Josiah Willard Gibbs (1839-1903), an American.Boltzman is credited with the initiation of statistical physics, and the prob-abilistic interpretation of entropy. His work paved the way for quantum theory.Boltzman was preceded by Maxwell who thought of molecules as elastic solids,whose behavior can be studied through the methods of probability.In 1902 Gibbs, who was occupied with problems of mechanics, published hisfamous book “
Basic Principles of Statistical Mechanics’ ’. This book was aninfluential development for the enhancement of probabilistic notions in physics.
Towards the beginning of the 20-th century, great inroads were made in proba-bility as a mathematical discipline by Chebychev, Markov, and Lyapunov, andinto its inroads in physics by Maxwell, Boltman, and Gibbs. However, mathe-maticians were repeatedly pointing out concerns regarding the need for a precisemeaning of probability.Indeed Bertrand (of Bertrand’s Paradox, of which is Borel’s Paradox, and thethree envelope problem are examples), and Henri Poincare, via their paradoxestried to emphasize the inaccuracies and vaguenesses in the basic notions ofinterpreting probability.Emil Borel (1871-1956) and Henri Poincare (1854-1912), both prominentFrench mathematicians, were determinists whose notion of probability, was thatit is a reflection of our ignorance. Both wrote two highly influential books onthe subject, and called for a rigorous definition of the meaning of probability.These can be seen as paving the path towards, Stage IV and V, on the axiomaticand the subjective development of probability.
The axiomatic method in science, particularly, the mathematical sciences, makesit possible to apply any theory to many areas. For example, Lobachevskii (1829)suggested the possibility of constructing geometry based on a system of axioms,different from those of Euclid, whereas Hilbert, Peano, and Kagan, investigatedsuch a possibility for geometry in the early part of the 20-th century; Hilbertand Peano also did this for arithmetic.With probability, Laplace’s classical definition using equiprobable events wasa tautology, because: equiprobable ⇔ equal probability. Also, the subjectiveinterpretation of probability had, at the early part of the 20-th century, seriousflaws having to do with a linear utility for money and state dependence. As aconsequence, the need for axiomatization was becoming more and more pressing.In 1917, S.N. Bernstein (1880-1968) published a paper hinting the axiomati-zation of probability. This marked a new stage in its development. Bernstein’saxiomatization was based on the notion of qualitative comparisons of events in10hich larger and smaller probabilities serve as a foundation. Bernstein’s ideaswere further developed by Glivenko and more recently, by Koopman (1940).Bernstein’s notion of probability was also materialistic, and was for applica-tions to the natural sciences.Richard von Mises (1883-1953) was a strong critic of both the equiprobableand the subjective theory of interpreting probability. His main contribution isthe frequency approach; i.e. probability is relevant only to mass phenomena.Approaches alternate to von Mises, were due to Keynes, followed by HaroldJeffreys, who viewed probability as a degree of likelihood, wherein every propo-sition has a certain definite probability. It is said that later on, Keynes recantedthis position.Simultaneous with attempts to lay the foundations of probability were rapidnew developments in the mathematical sciences, vis a vis the works of Khinchin,Borel, Cantelli, Hardy, Littlewood, and Hausdorff. These trends facilitated Kol-mogorov to construct his axiomatization of probability and lay the foundationfor a decisive stage in its development. In particular, Bernoulli’s result on theweak law and Borel’s on the strong law, led Kolmogorov to notice the connectionbetween probability and measure, and thus began his work on axiomatization,resulting in the publication of his famous book, in 1933.Kolmogorov’s aim was not to clarify the meaning of probability, but to estab-lish a branch of mathematics in exactly the same way as algebra and geometry.To Kolmogorov, the concept of a theory of probability is a system of sets whichsatisfy certain conditions. He thus introduced the term probability in the abovecontext, detached from any real world meaning.Not all applied scenarios satisfy Kolmogorov’s set up and architecture. Con-sequently, there are alternatives to probability like Zadeh’s Possibility Theoryfor fuzzy sets, and the Dempster-Shafer Belief Function Theory. Approaches at interpreting probability, alternate to the “classical” one of LaPlace, the “frequency” one of von Mises, the “logical” one of Keynes, as wellas the axiomatics of Kolmogorov (that technically speaking are free of inter-pretation) were due to de Finetti, and Ramsey, who interpret probability asa subjective quantity, personal to each individual. Whereas de Finetti inter-prets probability as a two-sided bet assuming a linear utility for money, Savage,motivated by Ramsey takes an axiomatic approach.Savage’s approach to personal probability was modeled after von-Neumannand Morgernstern’s axiomatic development of utility theory. This approach,is the most widely referenced approach to personal probability; it has as itsfoundation, behavioristic axioms of choice. Perhaps it is not too well recognizedthat these axioms appear to be rooted in Bernstein’s qualitative comparison ofevents; save for the feature that they pertain to choices between actions in theface of uncertainty.A striking feature of Savage’s axioms is that their consequences lead to thesimultaneous existence of both, a subjective probability and a utility, and the11aximization of expected utility as a recipe for decision making under uncer-tainty. Savage’s subjective probability conforms to the Kolmogorov axioms;however, in the latter’s set up, conditional probability is a definition, whereasin the former, it is a consequence of the Savage axioms.
Acknowledgements
The work reported here was supported by a grant from the City Universityof Hong Kong, Project Number 9380068, and by the Research Grants CouncilTheme-Based Research Scheme Grant T32-102/14N and T32-101/15R. Com-ments by Glen Shafer and Fabrizio Ruggeri have helped correct some inaccuracyin the orginal versions.
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