Fermat number

Fermat number inference - 2/2 Re:Stanford Challenge

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• published: 01 May 2007
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In this two-part set of clips, I take a look at a particular inference on the so-called Fermat numbers. These numbers were inferred to be only prime numbers, but that later turned out to be false. I show how to tackle this problem using statistical tools, in order to see exactly how much you would trust the inference in this case. I do not elaborate too much on the statistical methods, so this can be seen more as a demo than anything else. In this second clip, I study the Bayesian approach. I start off with a numeric experiment to determine the a' priori' probability that an arbitrary sequence produces only prime numbers. I use conservative estimate of 10% which produces an a' posteriori (after data) probability of 94%, with a 6% chance of being wrong. I then try a more sophisticated method, using a probability distribution for the prior probability, yielding a mean posterior probability of 19% for being wrong in the inference. Thus, the analysis shows that even though the data indicates that the sequence contains only prime numbers, we can not put a lot of trust in that assumption. When the next part of the sequence arrives, the probability for the sequence being only prime collapses to zero. Some useful wikipedia links: http://en.wikipedia.org/wiki/Bayes_theorem (Tells how to deduce Bayes theorem from the rules of probability) http://en.wikipedia.org/wiki/Bayesian_probability (Contains a description of probability distributions on probabilities!) Intro to Bayesian statistics with comparisons between Bayesian and frequentist statistics (somewhat biased, perhaps): http://www.cs.ubc.ca/~hutter/earg/presentations/whyBayes.ppt The number of primes below specified thresholds can be found here: http://primes.utm.edu/howmany.shtml One very useful book: E.T. Jaynes Probability theory - the logic of science