PeterDonis said:
That's not the relevant "prior" because Sleeping Beauty already knows on Sunday Night that she will not be awakened at all in the (H, H) case, so that case drops out of any calculation of probabilities conditioned on her being awakened. And she knows that the question she is going to be asked is about a probability conditioned on her being awakened (since she knows she will only get asked the question on being awakened).
One of the interesting aspects of Probability is that there can be many ways to create a sample space. If I roll two six-sided dice, I can create a sample space with 36 pair-combinations, or 11 sums. It's just easier to assign probabilities to the pair-combinations.
You can claim that "
a relevant prior" is the state of the coin(s) that SB can predict, on Sunday Night, will exist when she awakes. That's completely impractical, since it is also the posterior she needs to use. So of course there is no "new information."
But there is no such thing as "
the relevant sample space." You are using circular reasoning, by claiming I can only use the same space for both prior and posterior. And quite wrong.
My prior sample space, which I find to be more relevant than yours, is {(H,H), (H,T), (T,H), (H,H)}. The probability for each one-outcome event is 1/4.
- This is what the experiment moderator sees when he decides whether or not to wake SB.
- The resultant event space exists whether or not he wakes her.
- It continues to exist whether or not he wakes her.
- By waking SB, he does not create a sample space. He makes her a conditional observer of the one that exists.
- When she becomes an observer, she can deduce that (H,H) is no longer a possibility.
- THIS IS NEW INFORMATION.
Please note that SB knows, before being put to sleep, how each part of this process can go. She knows the coins will be tossed. Se knows that there are four possible combinations the moderator will look at. She knows what will happen in each contingency. She knows she will be asked a question in only three of the four.
PeterDonis said:
But she already knows that this will be the case on Sunday Night.
Yep. What has that to do with whether, in
my prior (which has the advantage that the probabilities are not controversial), (HH) was a possibility and now it isn't?
PeterDonis said:
And she also knows that she will not be awakened at all in this state, so she knows this state is irrelevant for computing probabilities conditional on her being awakened, which are the relevant probabilities for questions she will be asked when she is awakened.
But the point you seem to overlook is that the state exists whether or not she is made an observer.
So change it. Wake her before and after you reverse the dime (or on both Monday and Tuesday in the original). If both coins are showing Heads in my version (or the coin landed Heads and it is Tuesday), ask her whether she likes
Gone With the Wind more than
The Wizard of Oz. Otherwise, ask her for her credence in Heads. But now she has what I hope you would call "new information." The prior is the one I called relevant, and the posterior is the one you said was
the relevant prior.
She can say her confidence in Heads is 1/3.
But now, why does it matter if you lied, and had no intention of waking her if you already knew her preference about movies? The state that determines what type of - or even if - an observer she becomes is a valid prior.
PeterDonis said:
No, I'm asking you, because you made a claim about what a halfer would say. If you're not going to back up that claim with details, you should not have made it in the first place.
And I'm saying that, just like I can't understand why you want to call what is clearly as posterior "the relevant prior," I have never seen it justified. If you want to see an example of it, google for "
double halfers."