I'm just following the principal principle. If I can figure out what the objective chances are, so can Beauty, and she can set her credences accordingly. — Srap Tasmaner
That is of no use to her. When awakened, she doesn't know whether she is in an awake state that she should assign a probability of 1/2 to or 1/4 to. — Andrew M
Now I think that's wrong. There is no discounting. None of the 1/2's should be reduced to 1/4's. Monday is not 1/2:1/4 either. — Srap Tasmaner
No. I'm not sure how to formalize this — Srap Tasmaner
what an agent knows about the outcome of a particular fair coin toss. — Andrew M
Which in Beauty's case is zilch, isn't it? — Srap Tasmaner
I agree about your double-header example, but don't see the similarity to SB at all. Interviewing here clearly gives you information. — Srap Tasmaner
It is indeed true that Beauty has no evidence that she can use to distinguish Monday from Tuesday. This does not mean that such evidence does not exist, only that she does not have it.When Beauty is asked, "What is your credence that the coin landed heads?" she knows there's a chance the experiment is using the heads protocol, in which case this is her one and only interview, and a chance that it is using the tails protocol, in which case this may be her first interview, last, or one of many, depending. By stipulation, there is no evidence she can use to distinguish one interview from another; all she has to go on is her knowledge of the experiment's design.
when she is in an interview she knows that SOTAI is not happening — JeffJo
I have tossed the coin and given you a box.
What are the chances there's a red marble in the box? — Srap Tasmaner
Yeah that one was too easy. — Srap Tasmaner
This is incorrect. What happens on Tuesday&HEADS is a part of the HEADS protocol, so you excluded part of it. And you treat the various possibilities inconsistently.If I condition on ~(Tuesday & HEADS), I exclude neither the heads protocol nor the tails protocol, as neither included it.
The "help" I am trying to offer, is to get you to see that you have to separate both protocols into individual days. And you are right, it will be of no help to you if you refuse to see this, just like you won't address my "four volunteers" proof that the answer is 1/3.This helps me not at all.
Her sample space is the set of four possible single-day protocols: {Monday&TAILS, Tuesday&TAILS, Monday&HEADS, Tuesday&HEADS}. Each has a prior probability of 1/4 to apply to a random single day in the experiment, which is all that Beauty knows is happening. But because she kn0ows SOTAI is happening on that single day, she ca rule out one of those outcomes. — JeffJo
What happens on Tuesday&HEADS is a part of the HEADS protocol, so you excluded part of it. — JeffJo
A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.
The "help" I am trying to offer — JeffJo
you won't address my "four volunteers" proof that the answer is 1/3. — JeffJo
100 tosses, presumed result of 50 heads and 50 tails, 150 interviews.
If the Beauties all guess tails all the time, they will get 100 right out of their 150 answers.
That looks like a 2/3 success rate, right? But is it?
Out of the 100 tosses, they got 50 of them wrong. Looked at this way, that's a 50% success rate. — Srap Tasmaner
Here's the spontaneous version of guessing-SB:
Suppose I'm going to teach Andy & Michael a little about probability. I'm going to flip a coin a bunch of times, but before each flip, they each guess. When they're right they get an M&M, and when we're done we'll count the M&M's and stuff. Now suppose before one toss, Michael guesses "Heads! Heads heads heads heads heads!!!!" If the coin lands heads, do I give him 1 M&M or 6? — Srap Tasmaner
Yes, it is. The bolded text tells the lab techs what to do - or more accurately, what not to do - on both days. It defines two protocols for TAILS: interview Monday, interview Tuesday. It defines two protocols for HEADS: interview Monday, sleep Tuesday. Even if they send her home that day, that would still be SOTAI.What happens on Tuesday&HEADS is a part of the HEADS protocol, so you excluded part of it. — JeffJo
(a) No it isn't. From the OP:
A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends. — Srap Tasmaner
What matters is that there is a protocol on both days for both HEADS and TAILS. And that one of these four protocols is inconsistent with Beauty being interviewed. You keep treating the fact that she sleeps through a day as if that makes the day nonexistent,or that it is not something the lab techs have to have included in their protocol.The only thing that matters is one for heads and two for tails
?????(b) If it were part of the heads protocol, by eliminating it, you would be eliminating heads as an outcome. Simply being interviewed would tell you the coin landed tails.
Yep. Get two thousand volunteers. Order them randomly from #1 to #2,000. House #1 thru #1,0000 in the HEADS wing of your lab, and #1,001 thru #2,000 in the TAILS wing. Then flip your fair a coin.If that seems like a tendentious interpretation, consider what happens as you increase the number of tails interviews: whatever the ratio, that's your odds it was tails. Do a thousand tails interviews, and it's a near certainty -- according to thirders -- that a fair coin lands tails.
I toss fair coin twice. I ask for your credence that the first toss landed heads only on {HH, TH, TT}.
My question is this: do you think this is equivalent to SB? And why or why not? — Srap Tasmaner
Sorry, I'm not getting your experiment, or its equivalence to SB. — Srap Tasmaner
Here's a variation of the experiment. Suppose that for Tuesday and Heads Beauty is also awakened and interviewed. At every interview she is informed whether or not it is a Tuesday and Heads interview. She knows these rules prior to the experiment. Naturally if she is informed that it is Tuesday and Heads at the interview, she can conclude with certainty that she is in a state associated with heads.
However if Beauty is told that it is not Tuesday and Heads at her interview, should she condition on that information or not? — Andrew M
Mon Tue H 1/3 1/3 T 1/6 1/6
Mon Tue H 1/2 T 1/4 1/4
H T H 2 -1 T -2 1
In terms of your M&M example, if two people guess tails and they are correct, they both get an M&M. To reflect Sleeping Beauty, the experiment is set up such that only one person gets to guess when the outcome is heads. If the person conditions on the fact that they are getting to guess at all, then they will know that they are more likely to be in the tails track. — Andrew M
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