Probability Three Hundred Years of Controversy

Probability: Three Hundred Years of Controversy Harrison B. Prosper

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ProbabilityProbabilityThree Hundred Years of ControversyThree Hundred Years of Controversy

Harrison B. ProsperFlorida State University

10 June, 2005


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OutlineOutline

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 PrologThe Prolog


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A Long Time Ago In A Land Far AwayA 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.


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My Kingdom For A HorseMy 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.”


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My Kingdom For A Horse - IIMy 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.”


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To Be Or Not To BeTo Be Or Not To Be

Consider a series of identical experiments, each with two outcomes:

Red and White

What is the probability probability of the outcome Red ?

The solution is indeterminate without further assumptions.

Act 1Act 1


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Pascal’s WagerPascal’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.


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Pascal’s Wager – IIPascal’s Wager – II

Sometime later (1670), Pascal considered the following two hypotheses:H1 God existsH2 God does not exist

and the following two actions:A1 Lead a pious lifeA2 Lead a worldly life

and assigned utilities (negative losses), or gains, to each hypothesis/action pair.


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Pascal’s Wager – IIIPascal’s Wager – III

H1 H2A1 ∞ (eternal bliss!) (no worldly pleasures)

A2 + (worldly pleasures) +∞ (eternal damnation!)

+ (worldly pleasures)

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!


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The BernouillisThe 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.


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The Bernouillis – IIThe 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.


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The Bernouillis - IIIThe 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!

0)1(1Pr 2

nppp

ns

n Number of trialss Number of successesp Probability of a success at each trial

Act 2Act 2


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The ReverendThe 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 Chances

in which a special case of what became known as Bayes’ Theorem appeared.


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Genius Gone WildGenius 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.


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Genius Gone Wild – IIGenius Gone Wild – II

Laplace’s Rule of Succession This 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:

21)Pr(

nssuccess


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Genius Gone Wild – IIIGenius Gone Wild – III

Laplace’s result yields some implausible conclusions:

1. A coin is tossed once, n=1, and lands heads, s=1, Pr(heads) on next throw is 2/3!

2. 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!


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Genius Gone Wild – IVGenius 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.


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Intermission (circa 1900)Intermission (circa 1900)

Probability is Subjective Pascal, Fermat Leibnitz The Bernouillis Laplace, Poisson Lagrange, Gauss Boltzmann Maxwell Gibbs,….

Probability is Objective Cournot Ellis Boole Venn …

Ok! Which team would you rather join: the A team or the B team?

Act 3Act 3


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The (British) Empire Strikes Back …orThe (British) Empire Strikes Back …orKarl Pearson (1857 – 1936)

P.C. Mahalanobis (1893 – 1972)

R.A. Fisher (1890 – 1962)


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The Revenge of the Biologists!The Revenge of the Biologists!

R. A. Fisher Maximum likelihood Significance tests Goodness-of-fit Sampling distributions

Neyman/Pearson Unbiased estimators Confidence intervals Hypothesis tests

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!


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The Fantastic FourThe 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!

The EndThe 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)

Documents

Probability Three Hundred Years of Controversy