From: Eugene Salamin <gene_salamin@yahoo.com> It's beginning to look like we don't agree on what Bayesian analysis is. I'm saying that the posterior probability after n tosses updates and supersedes the posterior after k < n tosses.? The posterior provides everything that can be known about x, the probability that the coin comes up heads.? Bayes can't tell you if the coin is fair; that's for you to decide knowing the posterior.
? --? Gene
--so: your idea is this(?): initial ("prior") distribution of the coin-bias x is uniform on (0,1). After T+H coin tosses, the posterior is a beta(1+T,1+H) distribution on (0,1). One keeps tossing coins and as we do so the posterior keeps changing. If at any moment the CDF of the beta(1+T, 1+H) distribution at 1/2, is either less than K/2 or greater than 1-K/2, then we stop and declare "coin appears unfair, with confidence>=1-K." Is that right? If so, that was wrong. Why? Well it is good in the sense that if the coin were unfair, then this test would (with probability=1) ultimately indeed terminate and say so (regardless of the K with 0<K<1). But it is no good in the sense that if the coin genuinely were fair, then by the "law of iterated logarithm" with probability=1 an infinite number of days will come, during which |T-H| > sqrt(1.99 * N * lnlnN) where N=T+H. Therefore, it is guaranteed (with probability=1) this procedure will terminate and announce the coin is unfair. Regardless of the value of K with 0<K<1. Because (in Wolfram notation) Beta[ 1/2, N/2-Sqrt[N*0.49*Log[Log[N]]], N/2+Sqrt[N*0.49*Log[Log[N]]] ] goes to 0 when N-->infinity. Here is a table for N=10^k k | Beta value 1 | 0.00369675 2 | 1.73794*10^-30 3 | 9.6119*10^-302 4 | 2.1738412136*10^-3011 5 | 1.715030375*10^-30104 6 | 6.5603975*10^-301032 7 | 2.6446880*10^-3010302 8 | 2.344465*10^-30103002 9 | 6.6534*10^-301029999 In short: this procedure always claims coin is unfair, whether it is or not. After 10^9 tosses it typically will report enormous confidence a fair coin is unfair, like 0.9999...9 with about 300 million nines. The root cause of the problem here was the standard "repeated test banana peel" that I'd mentioned previously.