Russ, Very nice calculations. It would have taken me quite a while to figure it out. Thanks!
Tom, Interesting article. At the end though, I think the review's author misses the point of why Bayes theorem was so controversial amongst the 'frequents' (which suggests the book's author might have missed the point as well). The controversy occurred because many early statisticians wanted to believe in a truly probabilistic future, but believed in an already determined past -- basically what most people believe in. In that sense, there is a probability that you will pick the counterfeit coin before you make the choice, the 'a priori' probability (given a random choice) is .33. After you pick a coin, either you picked the conterfeit one or you did not, thus there is no probability worth discussing; the 'a posteriori' probability that you picked the counterfeit coin is either 1 or 0. Once the coin is in my hand, and no matter how many times I flip it, there will never be a 4/5ths chance that I picked the counterfeit coin. What would that even mean?!? Or so the anti-Bayes people argued: We can talk about our best guess as to the truth all day, but we are NOT talking about probability when we do so. <http://en.wikipedia.org/wiki/Ronald_Fisher>tried to deal with the problem of using the present to guess the past with his 'likelihood' formulae. Under some circumstances likelihood and Bayes theorem will come to the same number, but other times they will not. Likelihood calculations are still around, but not as popular as Bayes, because it is much harder to derive the formulae (and sometimes harder to gather the needed data). Fun fact: Researcher's reliance on Bayes formula was what lead Fisher to insist throughout his life that there was no evidence that smoking caused cancer. There is now evidence he would accept, but no data at the time allowed what he deemed to be the proper calculations. Eric P.S. For any stats people who might be reading, I have published on the problem of creating confidence intervals around correlations corrected for attenuation due to measurement error. If your population correlation is near 0, then the probability distribution for sample correlations is symmetric, and likelihood and Bayes will give you the same answer. As you approach 1 (or -1), the probability distributions becomes highly asymmetric, and likelihood and Bayes will give quite different answers. (Confession of mathematical inadequacies: I tackled the problem through simulation, not through derivation). On Sun, Aug 7, 2011 07:48 PM, Russ Abbott <[email protected]> wrote: >> >When I read that review it wasn't obvious to me how he got the result that he did for the counterfeit coin example. So I worked it out for myself--and after a bit of thinking about it got the same answer. If you're interested it's <http://cs.calstatela.edu/wiki/index.php/Bayes%27_theorem#Counterfeit_coin_example>. (Let me know if you think I made any mistakes.) The calculation is at the bottom of the Bayes Theorem page on my wiki. >> >> > >-- Russ Abbott >_____________________________________________> Professor, Computer Science > California State University, Los Angeles > > Google voice: 747-999-5105 > blog: <http://russabbott.blogspot.com/> > > > vita: <http://sites.google.com/site/russabbott/> >_____________________________________________ > > > > > > >>On Sun, Aug 7, 2011 at 2:41 PM, Tom Johnson <<#>> wrote: > > >> >A review of a new book that may be of interest. > >>--tom johnson > > > > > >The Mathematics of Changing Your Mind > > > > >By JOHN ALLEN PAULOS > > > > >Published: August 5, 2011 > >> > >>Sharon Bertsch McGrayne introduces Bayes’s theorem in her new book with a >>remark by John Maynard Keynes: “When the facts change, I change my opinion. >>What do you do, sir?” > > >> > >Bayes’s theorem, named after the 18th-century Presbyterian minister Thomas >Bayes, addresses this selfsame essential task: How should we modify our >beliefs in the light of additional information? Do we cling to old assumptions >long after they’ve become untenable, or abandon them too readily at the first >whisper of doubt? Bayesian reasoning promises to bring our views gradually >into line with reality and so has become an invaluable tool for scientists of >all sorts and, indeed, for anyone who wants, putting it grandiloquently, to >sync up with the universe. If you are not thinking like a Bayesian, perhaps >you should be.> > > > ><http://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?_r=1&ref=books>> > > > > > >============================================================ > >FRIAM Applied Complexity Group listserv > >Meets Fridays 9a-11:30 at cafe at St. John's College > >lectures, archives, unsubscribe, maps at <http://www.friam.org> > > > > > > ============================================================ >FRIAM Applied Complexity Group listserv >Meets Fridays 9a-11:30 at cafe at St. John's College >lectures, archives, unsubscribe, maps at http://www.friam.org > Eric Charles Professional Student and Assistant Professor of Psychology Penn State University Altoona, PA 16601
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