We do, we know that we are now in the experiment. In the case where you don't wake up on heads at all, have you learned new information upon waking? If yes, why don't you learn something anytime the relative probabilities differ?
That's not new information. You know that you will be in the experiment before the coin toss. Information does not flow acausally.
To contrast with Monty Hall, after the MC opens the first door, you do have new information: you glean information from the MC's choice, which he made based on his knowledge of where the prize is.
In this problem, the researchers take no action visible to you after the coin flip, and your memory is wiped before each new observation you can make. So your knowledge about the flip's outcome is exactly the same after it occurs as before: 50% chance.
If that doesn't count as new information, does waking up if you were to only wake on tails count as new information?
If yes, what's the justification for distinguishing the two?
What happens if you are told it's your first awakening as soon as you get up? Bayes theorem would imply you can't think the chances of heads stay the same before and after hearing this, so at least one must not be 50%.
I might be misunderstanding your conditions, but I would say no, waking up only on tails would not count, unless of course you got to make your choice after the researcher indicates they're about to put you back to sleep. Then you obviously know that the coin flip was tails! But in any case where you have just awoken after having had your memory wiped, and you have not been told anything by the researcher, you have been given no new info.
I think, if you are told it's your first awakening, then you still must assume 50% heads or tails: in both cases, you will always have a first awakening, so there is no new info there. However, since you will not be told this in your second awakening, if that occurs, you know immediately the coin must have come up tails.
>I think, if you are told it's your first awakening, then you still must assume 50% heads or tails: in both cases, you will always have a first awakening, so there is no new info there.
How is this not a direct violation of Bayes theorem? The probability of learning that it is the first awakening differs based on what the coin flip was, so it requires an update.
When you wake up, you don't know which awakening it is. You claim the probability of heads is 50%.
Now, you are told it is the first awakening. The probability of learning that if heads is twice as much as the probability of learning that if tails, so your Bayes factor is 2:1. You therefore cannot still believe that the probability of heads is 50%.
We do, we know that we are now in the experiment. In the case where you don't wake up on heads at all, have you learned new information upon waking? If yes, why don't you learn something anytime the relative probabilities differ?