Thursday, 9 December 2010

The search for randomness

Last night I attended the talk by Percy Diaconis on "The search for randomness" (his publications are listed here). I liked his approach, which focussed on understanding simple ideas in depth.

Diaconis discussed the notions of Newton determinacy, which is physics, versus randomness through examples of coin tossing, fair dice, darts, and shuffling cards. I particularly liked his discussion on the transition to, and onset of, randomness in a deterministic system, following the ideas of Poincaré. Poincaré used the method of arbitrary functions to bound probabilities by the integral of the derivative of the probability density function (PDF). See Keller (1986) on "The probability of heads".

Diaconis developed a (deterministic) coin tossing machine to analyze the dynamical bias in the coin toss. He also used simple ideas, such as variable-speed strobe lights and wrapping of dental floss, to count turns of a coin. The main result is that a coin rotates about 35-40 times/second, and the flight of a coin takes about 1/2 second. An important observation is that a coin spinning on edge is biased.

Regarding roulette, the Eudemonic Pie was mentioned. In Australia and the UK, this book was titled "The Newtonian Casino". I had the pleasure of having lunch with Doyne Farmer, mentioned in the book, when he visited Perth in 1992.

There was no mention of Jaynes, but Bayes was touched upon; the Bayesian point-of-view is that a PDF for a coin toss is simply the best guess for next toss. He stated that De Finetti, Machi, and Smith (1990) 'defined' probability in "Theory of Probability: A Critical Treatment". I should read this book.

Diaconis talked briefly about the quality of and sampling used in random number generators (RNGs) (see How we learned to cheat at online poker) and described how Gilbert, Shannon, and Reed began the mathematical study of card shuffling by introducing a good model for how people shuffle cards. He did not cover RANDU, but talked about the Numerical Recipes authors' bet, and how they went back to physics to generate random numbers (as used on the book's CD-ROM). Nor did he comment on Wolfram's cellular automata RNG which are deterministic and arise from trivial reproducible ICs. See also Hector Zenil's RNG demonstration.

Finally, he was very critical of "big models", something that resonated for me.

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