That’s not quite what I meant. It is not the experimenter’s thoughts that I am uncomfortable with- it is the collection of possible experimental outcomes.
I will try to illustrate with an example. Let us say that I toss a coin either (i) two times, or (ii) until it comes up heads. In the first case, the possible outcomes are HH, HT, TH, or TT; in the second case, they are H, TH, TTH, TTTH, TTTTH, etc. It isn’t obvious to me that a TH outcome has the same meaning in both cases. If, for instance, we were not talking about likelihood and instead decided to measure something else, e.g. the portion of tosses landing on heads, this wouldn’t be the case; in scenario (i), the expected portion of tosses landing on heads is 1⁄4 + .5/4 + .5/4 + 0⁄4 = .5, but in scenario (ii), it would be 1⁄2 + .5/4 + (1/3)/8 + .25/16 + … = log(2); i.e. a little under .7.
Just a note here: the fact that a dataset has the same likelihood function regardless of the procedure that produced it is actually NOT a trivial statement—the way I see it, it a somewhat deep result which follows from the optional stopping theorem and the fact that the likelihood function is bounded. Not trying to nitpick, just pointing out that there is something to think about here. According to my initial intuitions, this was actually rather surprising—I didn’t expect experimental results constructed using biased data (in the sense of non-fixed stopping time) to end up yielding unbiased results, even with full disclosure of all data.