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Anders_Sandberg comments on How Much Evidence Does It Take? - Less Wrong
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This also shows why independently replicated scientific experiments (more independent boxes) are more important than experiments with high p-values (boxes with better likeliehood ratios).
But the p-values go exponentially close to one with the size of the study. If you had three studies that used 11 boxes, vs. one with 33, you'd get exactly the same posterior probability for the ticket being a winner.
In other words, more experiments are exponentially more valuable than higher p-values, but higher p-values are exponentially cheaper.