And as far as thinking that N was the random variable in the second case, I had, I’d thought it through, and basically concluded that since no data at all was being hidden from us by experimenter B, and since A and B followed the same procedure, the probability that specific outcome would be published by B was the same as that of A
now, there is a partial caveat. One might say “but… What if B’s rule was to only publish the moment he had at least 70%?”
So one might think there’s more possible ways it could have come out like that for A then B, so that may affect things..
But no, and there’s two ways to look at it...
One is if you ask “what is the Specific data A has?”, ie, not just cures out of total trials, but the specific outcome of each patient, the likelihoods would then correspond to one specific outcome. And since the trials are (assumed) independant of each other given that the effectiveness of the medication is held fixed, rearanging the patients doesn’t alter the likelihood. So they can be rearranged to match B’s data without changing anything, even if we know B’s data must have the property that the 100th trial was the first one to get to 70% or above.
So, what happens here.. you know that no matter what answer you get to the question “what, precisely, was A’s data”, the produced likelihoods would then end up being identical to the likelihood produced from B’s data.
ie, by the principle of expectation of future belief has to equal present belief, you’ve gained same knowledge from both.
Another way to look at it is to consider that the only difference between the two, under these assumptions, would be a combinatorial coefficient. And that would be independant of the effectiveness itself given the numbers of how many cures vs non cures.
So that’ll cancel out when computing the likelihood ratios.
Maybe in private notes?
And as far as thinking that N was the random variable in the second case, I had, I’d thought it through, and basically concluded that since no data at all was being hidden from us by experimenter B, and since A and B followed the same procedure, the probability that specific outcome would be published by B was the same as that of A
now, there is a partial caveat. One might say “but… What if B’s rule was to only publish the moment he had at least 70%?”
So one might think there’s more possible ways it could have come out like that for A then B, so that may affect things..
But no, and there’s two ways to look at it...
One is if you ask “what is the Specific data A has?”, ie, not just cures out of total trials, but the specific outcome of each patient, the likelihoods would then correspond to one specific outcome. And since the trials are (assumed) independant of each other given that the effectiveness of the medication is held fixed, rearanging the patients doesn’t alter the likelihood. So they can be rearranged to match B’s data without changing anything, even if we know B’s data must have the property that the 100th trial was the first one to get to 70% or above.
So, what happens here.. you know that no matter what answer you get to the question “what, precisely, was A’s data”, the produced likelihoods would then end up being identical to the likelihood produced from B’s data.
ie, by the principle of expectation of future belief has to equal present belief, you’ve gained same knowledge from both.
Another way to look at it is to consider that the only difference between the two, under these assumptions, would be a combinatorial coefficient. And that would be independant of the effectiveness itself given the numbers of how many cures vs non cures.
So that’ll cancel out when computing the likelihood ratios.