The opinionator piece is misleading about the meaning of probability, especially as it implies that you could reformulate every problem into one about frequencies. As was pointed out a long time ago by Harold Jeffreys and repeated by Ed Jaynes,
"The essence of the present theory is that no probability, direct, prior, or posterior, is simply a frequency" -- H. Jeffreys (1939) "... a probability is a theoretical construct, on the epistemological level, which we assign in order to represent a state of knowledge, or that we calculate from other probabilities according to the rules of probability theory. A frequency is a property of the real world, on the ontological level, that we measure or estimate. So for us, probability theory is not an Oracle telling how the world must be; it is a mathematical tool for organizing, and ensuring the consistency of, our own reasoning." -- E.T. Jaynes[1] There are lots of simple problems you will not be able to reason correctly if you assume a probability is a frequency (e.g. Bernoulli's Urn). Not every event can be repeated a large number of times such that a frequency is obtained. This difficulty has often pushed "frequentists", to be confused between the result of measuring one thing many times and the result of measuring many things one time. For example, one could ask, by what basis do you draw a conclusion from the few that were surveyed to the millions that were not? Monitoring 1000 women and finding that 0.8% of them get breast cancer, tells you nothing about the health of the patient standing in front of you! Unless you have some other prior information linking the one to the one-thousand. What is really relevant would be to measure the one patient many times, but this is not always possible. I find that Jaynes book on Probability Theory[2] is THE definitive guide on how to teach it. For the less mathematically inclined, Chapter 1, which is freely available[3] online as a PDF is very readable and eye-opening. [1] Jaynes, E. T., 1989, `Clearing up Mysteries - The Original Goal, ' in Maximum-Entropy and Bayesian Methods, J. Skilling (ed.), Kluwer, Dordrecht, p. 1; [2] http://www.amazon.com/Probability-Theory-Logic-Science-Vol/dp/0521592712 [3] http://bayes.wustl.edu/etj/prob/book.pdf -----Original Message----- From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On Behalf Of Colin Nave Sent: April-27-10 4:09 AM To: CCP4BB@JISCMAIL.AC.UK Subject: Re: [ccp4bb] Help with Bayes's theorem This is quite a good one as well http://www.inference.phy.cam.ac.uk/mackay/pope.html I recall the pope analysis followed on from a Nature article covering the O J Simpson example (also covered in the NY times link). Sean Eddy who is an author on the above link wrote what I regard as an excellent intro to Bayesian statistics ftp://selab.janelia.org/pub/publications/Eddy-ATG3/Eddy-ATG3-reprint.pdf Colin > -----Original Message----- > From: CCP4 bulletin board [mailto:ccp...@jiscmail.ac.uk] On > Behalf Of Jim Pflugrath > Sent: 26 April 2010 20:30 > To: CCP4BB@JISCMAIL.AC.UK > Subject: [ccp4bb] Help with Bayes's theorem > > I thought some of you would enjoy a little conditional > probability discussion found in the NY Times today, since > this is a big part of crystallography nowadays. I'm always > on the lookout for good ways to teach Bayes's theorem. > > http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/ > > Jim >
