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Probability Three Hundred Years of Controversy

Probability Three Hundred Years of Controversy. Harrison B. Prosper Florida State University 10 June, 2005. Outline. The Prolog Act 1 Pascal’s Wager The Bernouillis Act 2 The Reverend Genius Gone Wild Act 3 The Empire Strikes Back The Fantastic Four. The Prolog.

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Probability Three Hundred Years of Controversy

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  1. ProbabilityThree Hundred Years of Controversy Harrison B. Prosper Florida State University 10 June, 2005 Probability: Three Hundred Years of Controversy Harrison B. Prosper

  2. Outline • The Prolog • Act 1 • Pascal’s Wager • The Bernouillis • Act 2 • The Reverend • Genius Gone Wild • Act 3 • The Empire Strikes Back • The Fantastic Four Probability: Three Hundred Years of Controversy Harrison B. Prosper

  3. The Prolog

  4. A Long Time Ago In A Land Far Away • It was recognized that: • Some things happen by chance. • Order can arise out of chaos. • But, although gambling was common in these ancient lands, there is no evidence that a theory of chance existed before the 17th Century, …. with perhaps one exception. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  5. My Kingdom For A Horse The Tale of King Nala (from Mahabharata), or, you lose some, you win some! “King Nala lost his kingdom in a gambling contest and ended up working for King Bhangasuri as a chariot-driver.” “One day, while on a journey with the king, Nala boasted of his mastery of horses.” Probability: Three Hundred Years of Controversy Harrison B. Prosper

  6. My Kingdom For A Horse - II “King Bhangasuri was annoyed and reminded Nala that no man knows everything. And to make his point the king made a quick estimate of the number of fruit on a nearby tree.” “Nala counted the fruit and was amazed by the accuracy of the king’s estimate. When Nala asked the king how he did this, the latter replied: Know that I am a knower of the secret of the dice and therefore adept in the art of enumeration.” Probability: Three Hundred Years of Controversy Harrison B. Prosper

  7. To Be Or Not To Be Consider a series of identical experiments, each with two outcomes: Red and White What is the probability of the outcome Red ? The solution is indeterminate without further assumptions. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  8. Act 1

  9. Pascal’s Wager In 1654, the gambler, Chevalier de Méré, observed that his two ways of betting did not work equally well and complained to Blaise Pascal that the rules of arithmetic must surely be faulty! The problem was to work out which of two kinds of bet was better: 4 throws of a die to get at least one 6, or 24 throws to get a “double 6”. Pascal’s answer: 4 throws is slightly better with a chance of 1 – (5/6)4. Thus was (re-)born the mathematical theory of probability. The first work was by the Italian Gerolamo Cardano in 1564. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  10. Pascal’s Wager – II Sometime later (1670), Pascal considered the following two hypotheses: H1 God exists H2 God does not exist and the following two actions: A1 Lead a pious life A2 Lead a worldly life and assigned utilities (negative losses), or gains, to each hypothesis/action pair. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  11. Pascal’s Wager – III Pascal argued that if your prior Prob(H1) > 0, however small, then your expected gain from being pious > expected gain from being worldly. So if you believe in God, even if only on Sundays, it is in your interest to live a saintly life! Probability: Three Hundred Years of Controversy Harrison B. Prosper

  12. The Bernouillis In the late 17th Century, probability was viewed in many ways: • as something related to the fraction of favorable outcomes in a set of outcomes considered equally likely, • as something related to uncertain knowledge of outcomes, • as a physical tendency in things that exhibit chance. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  13. The Bernouillis – II James Bernouilli (1654 – 1705)labored hard to make sense of these ideas. Alas, he was dissatisfied with his efforts and chose not to publish them. However, Bernouilli’s famous Ars Conjectandi was published later by his nephew Nicholas Bernouilli in 1713. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  14. The Bernouillis - III One of Bernouilli’s important results is the so-called Weak Law of Large Numbers The meaning of this result is still hotly debated. The difficulty is that the probability p is defined in terms of another probability, namely, Pr! n Number of trials s Number of successes p Probability of a success at each trial Probability: Three Hundred Years of Controversy Harrison B. Prosper

  15. Act 2

  16. The Reverend Thomas Bayes (1702 – 1761) read (via a proxy!) the following paper before the Royal Society, on 23 December, 1763: An Essay towards solving a Problem in the Doctrine of Chancesin which a special case of what became known as Bayes’ Theorem appeared. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  17. Genius Gone Wild Pierre Simon de Laplace (1749 – 1827) published a book (in 1812) entitled: Théorie Analytique des Probabilitiés In it, the general form of Bayes’ Theorem is proved and probability theory is applied to many problems. But some of hisresults soon became controversial. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  18. Genius Gone Wild – II Laplace’s Rule of SuccessionThis is a special case of his general result for the probability of a success on the next trial, Pr(success), given that s successes have occurred in n trials: Probability: Three Hundred Years of Controversy Harrison B. Prosper

  19. Genius Gone Wild – III Laplace’s result yields some implausible conclusions: • A coin is tossed once, n=1, and lands heads, s=1, Pr(heads) on next throw is 2/3! • A boy succeeds s=8 times to reach his birthday after trying 8 times. His chance to live to 9 is 9/10! But a 98 year-old has a 99/100 chance of making it to 99! Probability: Three Hundred Years of Controversy Harrison B. Prosper

  20. Genius Gone Wild – IV These results and the whole idea of subjective probability were ridiculed severely by some philosophers, especially by the Englishman John Venn (1866). However, instead of throwing the baby out with the bathwater, Venn et al. ought merely to have thrown out the flat prior used by Laplace and replace it with something better. Probability: Three Hundred Years of Controversy Harrison B. Prosper

  21. Probability is Subjective Pascal, Fermat Leibnitz The Bernouillis Laplace, Poisson Lagrange, Gauss Boltzmann Maxwell Gibbs,…. Probability is Objective Cournot Ellis Boole Venn … Intermission (circa 1900) Ok! Which team would you rather join: the A team or the B team? Probability: Three Hundred Years of Controversy Harrison B. Prosper

  22. Act 3

  23. Karl Pearson (1857 – 1936) R.A. Fisher (1890 – 1962) P.C. Mahalanobis (1893 – 1972) The (British) Empire Strikes Back …or Probability: Three Hundred Years of Controversy Harrison B. Prosper

  24. R. A. Fisher Maximum likelihood Significance tests Goodness-of-fit Sampling distributions Neyman/Pearson Unbiased estimators Confidence intervals Hypothesis tests The Revenge of the Biologists! Each school was brutally critical of the other. Fisher was particularly scornful of confidence intervals, noting that science and mass production are not precisely the same!The only thing they agreed on was that Laplace and Co. were clearly mystical dimwits! Probability: Three Hundred Years of Controversy Harrison B. Prosper

  25. The Fantastic Four • As a result of Fisher’s onslaught his views, and in spite of his best efforts those of Neyman, prevailed. • Oddly, most physicists today (claim) adherence to Neyman’s mass production view of inference: Produce results so that over your career 68% of them will be correct. • However, starting in the 1950s, the views of the Bernouillis, Bayes and Laplace have enjoyed a significant comeback, and even has a toe-hold amongst CDF and DØ physicists! Probability: Three Hundred Years of Controversy Harrison B. Prosper

  26. The End Theory of Probability, 3rd Edition, Sir Harold Jeffreys, (Oxford, 1961) Probability: The Logic Of Science, Edwin Jaynes (2001) Statistical Thought: A Perspective and History, S. K. Chatterjee (Oxford, 2003) How Probabilities Came to Be Objective and Subjective, Loraine Daston, Hist. Math. 21 330-344 (1994) Probabilities Are Single-Case, Or Nothing, D.M. Appleby,quant-ph/0408058 v1 (2004)

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