Paul_Gowder comments on Beautiful Probability - Less Wrong
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I have to say, the reason the example is convincing is because of its artificiality. I don't know many old-school frequentists (though I suppose I'm a frequentist myself, at least so far as I'm still really nervous about the whole priors business -- but not quite so hard as all that), but I doubt that, presented with a stark case like the one above, they'd say the results would come out differently. For one thing, how would the math change?
But the case would never come up -- that's the thing. It's empty counterfactual analysis. Nobody who is following a stopping rule as ridiculous as the one offered would be able to otherwise conduct the research properly. I mean, seriously. I think Benquo nailed it: the second researcher's stopping rule ought to rather severely change our subjective probability in his/her having used a random sample, or for that matter not committed any number of other research sins, perhaps unconsciously. And that in turn should make us less confident about the results.