Tag Archives: George Polya

Historians and probability

Dice players

Dice players (detail, Georges de La Tour, 17th C.)

Bayesian probability theory is a formal method of reasoning about evidence. Its probabilities are typically subjective and personal measures. They represent either a real person’s felt confidence, or a hypothetical person’s theoretically justified confidence. Please do not be put off by the word subjective. Justified confidence is the foundation of prudent belief, action and behavior.

Richard Carrier is a serious independent scholar and internet celebrity who earned his doctorate in ancient history from Columbia University. He uses Bayesian methods to study history, especially the question of whether Jesus was a real historical person. Carrier professes serene assurance about the objectivity and validity of his Bayesian approach to history (link),

“I don’t think I’ll convince everyone, but the only people who won’t be convinced are people who are irrationally, dogmatically opposed to what I’m arguing.”

This post discusses how well Bayesian methods can resolve historical controversies, in the sense of achieving consensus founded on objective analysis of evidence. Within a community of Bayesians, objectivity and near-unanimity aren’t completely out of reach, but they tend to be elusive except when most people would be convinced whether or not they appeal to Bayes.

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De Finetti’s conjecture: First broken, then fixed; but nobody noticed, Part 2

In the first part, we left Bruno de Finetti in 1949 as he established that for four distinct individual possibilities (like which team will win a championship), any usual ordering of “tickets” that was “quasi-additive” was also “probabilistic.” He conjectured that this would be true for any finite number of quasi-additively ordered propositions, and invited the community to help prove him right or wrong.

One who accepted de Finetti’s invitation was Leonard Savage, who later developed his own landmark axiomatization of subjective probability. Savage gave the obscure 1949 paper to Charles Kraft, John Pratt and Abraham Seidenberg. They showed in 1959 that de Finetti’s conjecture was wrong if there are five or more basic outcomes.

That’s the “broken” part of the story. De Finetti’s conjecture, scribbled in haste to rebut his friend George Polya, is false. Many people scrambled to fix it. The first were Kraft, Pratt and Seidenberg themselves in 1959. A famous second time was five years later, by Dana Scott. Oddly, neither solution was satisfactory to its authors. Odder still was that De Finetti himself may have come within a whisker of repairing it back at the beginning in 1949.  Continue reading

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De Finetti’s conjecture: First broken, then fixed; but nobody noticed, Part 1

Bruno de Finetti was an Italian statistician and probability theorist who contributed mightily to the Twentieth Century revival of Bayesian methods in statistics. One of his achievements was a gambling semantics for probability which explained Laplace’s observation that probability seemed to measure a person’s “degrees of belief” about uncertain prospects.

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