Suppose that the frequency of cures actually converges at 60%, and each researcher performs his experiment 100 times. Researcher A should end up with about 6000 cures out of 10000, and Researcher B should end up with .7n cures out of n. We would expect in advance, after being told that Researcher B ended up testing 10000 people, that he encountered 7000 cures.
It seems that once we know that Researcher B was not going to stop until the incidence of cures was 70%, we do not learn anything further about the efficacy of the treatment by looking at his data.
You are proposing to partition the data into two observations, namely “number of trials that were performed” and “the results of those trials”.
The information you have after observing the first part and not the second, does depend on the researcher’s stopping criterion. And the amount of information you gain from the second observation also so depends (since some of it was already known from the first observation in one case and not in the other). But your state of information after both observations does not depend on the stopping criterion.
Also, the most likely outcome of your proposed experiment is not .7n cures for some n. Rather, it’s that Researcher B never stops.
Suppose that the frequency of cures actually converges at 60%, and each researcher performs his experiment 100 times. Researcher A should end up with about 6000 cures out of 10000, and Researcher B should end up with .7n cures out of n. We would expect in advance, after being told that Researcher B ended up testing 10000 people, that he encountered 7000 cures.
It seems that once we know that Researcher B was not going to stop until the incidence of cures was 70%, we do not learn anything further about the efficacy of the treatment by looking at his data.
What’s wrong about this?
You are proposing to partition the data into two observations, namely “number of trials that were performed” and “the results of those trials”.
The information you have after observing the first part and not the second, does depend on the researcher’s stopping criterion. And the amount of information you gain from the second observation also so depends (since some of it was already known from the first observation in one case and not in the other). But your state of information after both observations does not depend on the stopping criterion.
Also, the most likely outcome of your proposed experiment is not .7n cures for some n. Rather, it’s that Researcher B never stops.
That explanation sounds plausible. Can you say it again in math?