A better and simpler chisquare test
The idea is so simple that surely someone must have imagine it before. Just an example suffice: consider the experiment of throwing a dice, this can be seen as five different independent experiments counting the numbers in the remaining cases and applying the normal approximation to the binomial, so that for example in a dice the probabilies are 1/6, 1/5, 1/4 .... and you sum those (oi - restpi)/(rest*pi*qi) where pi are the updated probabilies and rest are the remaining number in each case. The sum follows approximately a n-1 dof chisquare. In the examples in wikipedia the numbers are very different for the first example (good dice) but similar for the second (biased dice) Bayesian intuition or common sense implies that the n-1 experiment are independent.