Well, right, assuming we’re Bayesians, but when we’re just “rejecting the null hypothesis” we should mostly be concerned about likelihood from the null hypothesis which has no moving parts, which is why I used the log approximation I did. But at this point we’re mixing frequentism and Bayes to the point where I shan’t defend the point further—it’s certainly true that once a Bayesian considers more than exactly two atomic hypotheses, the update on two independent pieces of evidence doesn’t go as the square of one update (even though the likelihood ratios still go as the square, etc.).
Well, right, assuming we’re Bayesians, but when we’re just “rejecting the null hypothesis” we should mostly be concerned about likelihood from the null hypothesis which has no moving parts, which is why I used the log approximation I did. But at this point we’re mixing frequentism and Bayes to the point where I shan’t defend the point further—it’s certainly true that once a Bayesian considers more than exactly two atomic hypotheses, the update on two independent pieces of evidence doesn’t go as the square of one update (even though the likelihood ratios still go as the square, etc.).