Split the researchers that generate the data from the reasoner who is trying to estimate the hidden double from the data.
What is the data that the estimator receives? There is clearly a string of 100 bits indicating the results of the comparisons, but there is also another datum which indicates that the experiment was stopped after 100 iterations. This is a piece of evidence which must be included in the model, and the way to include it depends on the estimator’s knowledge of the stopping criterion used by the data generator.
The estimator has to take into account the possibility of cherry picking.
EDIT:
I think I can use an example:
Suppose that I give you N =~ 10^9 bits of data generated according to the process you describe, and I declare that I had precommitted to stop gathering data after exactly N bits. If you trust me, then you must believe that you have an extremely accurate estimate of the hidden double. After all, you are using 1 gigabit of data to estimate less than 64 bits of entropy!
But then you learn that I lied about the stopping criterion, and I had in fact precommitted to stop gathering data at the point that it would have fooled you into believing with very high probability that the hidden number was, say, 0.42.
Should you update your belief on the hidden double after hearing of my deception? Obviously you should. In fact, the observation that I gave you so much data now makes the estimate extremely suspect, since the more data I give you the more I can manipulate your estimate.
So, suppose I know the stopping criterion and the number of button presses that it took to stop the sequence, but I wasn’t given the actual sequence.
It seems to me like I can use the two of those to recreate the sequence, for a broad class of stopping criteria. “If it took 100 presses, then clearly it must be 70 1s and 30 0s, because if it had been 71 1s and 29 0s he would have stopped then and there would be only 99 presses, but he wouldn’t have stopped at 69 1s and 30 0s.” I don’t think I have any additional info.
Should you update your belief on the hidden double after hearing of my deception? Obviously you should.
Update it to what? Assuming that the data is not tampered with, just that your stopping criterion was pointed at a particular outcome, it seems like that unless the double is actually very close to 0.42 then you are very unlikely to ever stop!* It looks like the different stopping criteria impose conditions on the order of the dataset, but the order is independent of the process that generates whether each bit is a 1 or a 0, and thus should be independent of my estimate of the hidden double.
* If you imagine multiple researchers, each of which get different sequences, and I only hear from some of the researchers- then, yes, it seems like selection bias is a problem. But the specific scenario under consideration is two researchers with identical experimental results drawing different inferences from those results, which is different from two researchers with differing experimental setups having different distributions of possible results.
Split the researchers that generate the data from the reasoner who is trying to estimate the hidden double from the data.
What is the data that the estimator receives? There is clearly a string of 100 bits indicating the results of the comparisons, but there is also another datum which indicates that the experiment was stopped after 100 iterations. This is a piece of evidence which must be included in the model, and the way to include it depends on the estimator’s knowledge of the stopping criterion used by the data generator.
The estimator has to take into account the possibility of cherry picking.
EDIT:
I think I can use an example:
Suppose that I give you N =~ 10^9 bits of data generated according to the process you describe, and I declare that I had precommitted to stop gathering data after exactly N bits. If you trust me, then you must believe that you have an extremely accurate estimate of the hidden double. After all, you are using 1 gigabit of data to estimate less than 64 bits of entropy!
But then you learn that I lied about the stopping criterion, and I had in fact precommitted to stop gathering data at the point that it would have fooled you into believing with very high probability that the hidden number was, say, 0.42.
Should you update your belief on the hidden double after hearing of my deception? Obviously you should. In fact, the observation that I gave you so much data now makes the estimate extremely suspect, since the more data I give you the more I can manipulate your estimate.
So, suppose I know the stopping criterion and the number of button presses that it took to stop the sequence, but I wasn’t given the actual sequence.
It seems to me like I can use the two of those to recreate the sequence, for a broad class of stopping criteria. “If it took 100 presses, then clearly it must be 70 1s and 30 0s, because if it had been 71 1s and 29 0s he would have stopped then and there would be only 99 presses, but he wouldn’t have stopped at 69 1s and 30 0s.” I don’t think I have any additional info.
Update it to what? Assuming that the data is not tampered with, just that your stopping criterion was pointed at a particular outcome, it seems like that unless the double is actually very close to 0.42 then you are very unlikely to ever stop!* It looks like the different stopping criteria impose conditions on the order of the dataset, but the order is independent of the process that generates whether each bit is a 1 or a 0, and thus should be independent of my estimate of the hidden double.
* If you imagine multiple researchers, each of which get different sequences, and I only hear from some of the researchers- then, yes, it seems like selection bias is a problem. But the specific scenario under consideration is two researchers with identical experimental results drawing different inferences from those results, which is different from two researchers with differing experimental setups having different distributions of possible results.