This paper starts out with a misrepresentation. "As a reminder, this is the Sleeping Beauty problem:"... and then it proceeds to describe the problem as Adam Elga modified it to enable his thirder solution. The actual problem that Elga presented was:
Some researchers are going to put you to sleep. During the two days[1] that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking.2 When you are first awakened[2], to what degree ought you believe that the outcome of the coin toss is Heads?
There are two hints of the details Elga will add, but these hints do not impact the problem as stated. At [1], Elga suggests that the two potential wakings occur on different days; all that is really important is that they happen at different times. At [2], the ambiguous "first awakened" clause is added. It could mean that SB is only asked the first time she is awakened; but that renders the controversy moot. With Elga's modifications, only asking on the first awakening is telling SB that it is Monday. He appears to mean "before we reveal some information," which is how Elga eliminates one of the three possible events he uses.
Elga's implementation of this problem was to always wake SB on Monday, and only wake her on Tuesday if the coin result was Tails. After she answers the question, Elga then reveals either that it is Monday, or that the coin landed on Tails. Elga also included DAY=Monday or DAY=Tuesday as a random variable, which creates the underlying controversy. If that is proper, the answer is 1/3. If, as Neal argues, it is indexical information, it cannot be used this way and the answer is 1/2.
So the controversy was created by Elga's implementation. And it was unnecessary. There is another implementation of the same problem that does not rely on indexicals.
Once SB is told the details of the experiment and put to sleep, we flip two coins: call then C1 and C2. Then we perform this procedure:
After these steps are concluded, whether it happened in step 1 or step 3, we turn coin C2 over to show the opposite side. And then repeat the same procedure.
SB will thus be wakened once if coin C1 landed on Heads, and twice if Tails. Either way, she will not recall another waking. But that does not matter. She knows all of the details that apply to the current waking. Going into step 1, there were four possible, equally-likely combinations of (C1,C2); specifically, (H,H), (H,T), (T,H), and (T,T). But since she is awake, she knows that (H,H) was eliminated in step 1. In only one of the remaining, still equally-likely combinations, did coin C1 land on Heads.
The answer is 1/3. No indexical information was used to determine this. No reference the other potential waking, whether it occurs before or after this one, is needed. This implements Elga's question exactly; this only possible issue that remains is if Elga's implementation does.
The "need to throw the second coin" is to make the circumstances underlying any awakening the same. Using a random method is absolutely necessary, although it doesn't have to be flipped. The director could say that she is choosing her favorite coin face. As long as SB has no reason to think that is more likely to be one result than the other, it works. The reason Elga's version is debated, is because it essentially flips Tails first for the second coin.
What the coins are showing at the moment are elementary outcomes of the experiment-within-the-experiment. "Causal connection," whatever you think that means, has nothing to do with it since we are talking about a fixed state of the coins from the examination to the question.
Which statement here to you disagree with?
But there are other ways, that have whatever "causal connection" you think is important. That is, they match the way Elga modified the experiment. That's another point you seem to ignore, that the two-day version differs from the actual question by more than mine does.
Try using four volunteers instead of one, but one coin for all. Each will be wakened on both Monday and Tuesday, except:
This way, on each of Monday and Tuesday, exactly three volunteers will be wakened. And unless you think the specific days or coin faces have differing qualities, the same "causal connections" exist for all.
Each volunteer will be asked for her credence that the coin landed on the result where she would be wakened only once. With the knowledge of this procedure, an awake volunteer knows that she is one of exactly three, and that their credence/probability cannot be different due to symmetry. Since the issue "the coin landed on the result where you would be wakened only once" applies to only one of these three, this credence is 1/3.
AINC: "What's Beauty credence for Heads when she wakes on Wednesday and doesn't remember any of her awakenings on Monday/Tuesday?"
If she has no reason to think this is not one of her awakenings on Monday/Tuesday, then her credence is the same as it would have been then: 1/2 if she is a halfer, and 1/3 if she is a thirder.
AINC: "If it's 1/2 what is the reason for the change from 1/3?"
The only way it could change from 1/3 to 1/2, is if she is a thirder and you tell her that it is Wednesday. And the reason it changes is that you changes the state of her information, not because anything about the coin itself has changed.
But if you think her credence should be based on the actual day, even when you didn't tell her that is was Wednesday, then you have told an implicit lie. You are asking her to formulate a credence based on Monday/Tuesday, but expecting her answer to be consistent with Wednesday.
AINC: The point of the Wednesday question is to highlight, that, what you mean by "credence", isn't actually a probability estimate that the coin is Heads.
And the point of my answer, is that it is actually a conditional probability based on an unusual state of information.
My point is that SB must have reason to think that she exists in the "Monday or Tuesday" waking schedule, for her to assign a credence to Heads based on that schedule. If she is awake, but has any reason to think she is not in that situation, her credence must take that into account.
You told her that she would be asked for her credence on Monday and maybe on Tuesday. "What's Beauty credence for Heads when she wakes on Wednesday and doesn't remember any of her awakenings on Monday/Tuesday?" is irrelevant because you are allowing for the case where that is not true yet you want her to believe it is. That is lying to her, in the context of the information you want her to use.
But she can from an opinion. "My credence should be halfer/thirder answer if Wednesday has not yet dawned, or 1/2 if it has. Since the cue I was told would happen - being asked for my credence - has not yet occurred, I am uncertain which and so can't give a more definitive answer." And if you give that cue on Wednesday, you are lying even if you promised you wouldn't.
And yes, my mathematical model corresponds to SB's reality when she is asked for her credence. That is the entire point. If you think otherwise, I'd love to hear an explanation instead of a dissertation that does not apply.
Whether your knowledge correctly represents the state of the system, or not, is irrelevant. Your credence is based on your knowledge. If they lie to SB and wake her twice after Heads, and three times after Tails? But still tell here it is once or twice? A thirder's credence should still be 1/3.
And those who guess would not be considered the rational probability agent that some versions insist SB must be.
The correct answer is 1/3. See my answer to this question.
Ape in the coat: “What's Beauty credence for Heads when she wakes on Wednesday and doesn't remember any of her awakenings on Monday/Tuesday?”
The snarky answer is “irrelevant.” The assessment of her credence, that she gives on Monday or Tuesday, is based on the information that it is either Monday or Tuesday within the waking schedule described in the experiment. She is not responsible for incorporating false information into her credence.
One thing that seems to get lost in many threads about this problem, is that probability and/or credence is a function of your knowledge about the state of the system. It is not a property of the system itself. You are trying to make it a property of the system.
Halfers are most guilty of not understanding this, since their arguments are based on there being no change to the system despite an obvious change to the knowledge (I’m ignoring how to treat that knowledge). Thirders recognize the change, but keep trying to cast the difference in terms of episiotomy when it really is much simpler.
“However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1/3. Should she expect to successfully guess Tails with 2/3 probability?”
No, since her knowledge is different now than it was then.
“Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn't know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?”
If I understand this correctly, she knows that it is Wednesday. Her credence is 1/2. She can, at the same time on Sunday or Wednesday, make an assessment of the state of her knowledge on Monday or Tuesday, and recognize how it is different. In other words, "my credence now is 1/2, but my credence then should be 1/3." The paradox you seem to be fishing for does not exist.
No, much of the debate gets obscured by trying to ignore how SB's information, while it includes all of the information that will be used it separate the experiment into two observations, also is limited to the "inside information" and she is in just one of those parts. That's the purpose of the shopping spree I suggested. Correcting what you wrote, if the memory wipe is complete and SB has literally no way of knowin what information might apply to what is clearly a distinct part of the experiment, but she knows there is a different part than the current one, how can it not be "considered (by her) to be a different experiences."
For the purposes of sampling, there are two parts to the experiment. What you call them is not relevant, only that SB knows that she is in one, and only one, of two parts. In the classic version, as Elga modifided it from the actual problem he proposed, one part must have a waking and one part has either a sleeping or a waking. Are you really claiming that the part where she is left asleep is not a part of the experiment? One that she knows is a possibility that is ruled out by her current state? And so fits the classic definition of "new evidence" that you deny? But if you really believe that, what about my version where she is taken shopping? It is even better as a classic example of evidence.
Or you could look at my answer. There, I explain how the problem you are solving actually is an "alternate formulation." And, while there are still two parts, they are equivalent so we do not need to treat them separately. The answer is unequivocally 1/3.
Or try this: Instead of a coin, roll two six-sided dice. On Monday, ask her for the probability that the resulting sum is 7. On Tuesday, if the sum is odd, ask her the same question. But if it is even, ask her for the probability that the sum is 8.
If the room has a calendar, Beauty should certainly say Pr(S=7)=1/6 on Sunday and Monday. But on Tuesday, in answer to "what is the probability of 7," she should say Pr(S=7)=1/3. But without a calendar, Pr(S=7)=1/6 can't be right. Because it might be Tuesday, and Pr(S=7)=1/3 can't be right. And it might be Monday, when Pr(S=7)=1/6 can't be right. It has to be something in between, and that something is (2/3)*(1/6)+(1/3)*(1/3)=4/18=2/9. Yes, even though this is never the answer if she knows the day. That is how conditional probability works.
The "valid answers for different questions" position is a red herring, and I'll show why. But first, here is an unorthodox solution. I don't want to get into defending whether its logic is valid; I'm just laying some groundwork.
On Sunday (in Elga's re-framing of the problem he posed - see my answer), the sample space for the experiment seems to be {T,H} with equal probability for each outcome. Both Monday and Tuesday will happen, and so cannot be used to specify outcomes. This is one of your "different questions."
But when SB is awake, only one of those days is "current." (Yes, I know some debate whether days can be used this way - it is a self-fulfilling argument. If it is assumed to be true, you can show that it is true.) Elga was trying to use the day as a valid descriptor of disjoint outcomes, so that in SB's world the outcomes T&Mon and T&Tue become distinct. The prior probabilities of the three disjoint outcomes {T&Mon, T&Tue, H} are each 1/2, and the probability for H can be updated by the conventional definition:
Pr(H|T&Mon or T&Tue or H) = Pr(H)/[Pr(T&Mon)+Pr(T&Tue)+Pr(H)]
Pr(H|T&Mon or T&Tue or H) = (1/2)/[(1/2)+(1/2)+(1/2)] = 1/3
If you think about it, this is exactly what Elga argued, except that he didn't take the extreme step of creating a probability that was greater than 1 in the denominator. He made two problems, that each eliminated an additional outcome. I suggest, but don't want to get into defending, that this is valid since SB's situation divides the prior outcome T into two that are disjoint. Each still has the prior probability of the parent outcome.
But this is predicated on that split. Which is what I want to show is valid. What if, instead of leaving SB asleep on Tuesday after Heads, wake her but don't interview her? Instead, we do something quite different, like take her on a $5,000 shopping spree at Saks Fifth Avenue? (She does deserve the chance of compensation, after all). This gives her a clear way to distinguish H&Mon&Interview from H&Tue&Saks as disjoint outcomes. The use of the day as a descriptor is valid, since this version of the experiment allows for it to be observed.
The actual sample space, on Sunday, for what can happen in an "awake" world for SB is {T&Mon, H&Mon, T&Tue, H&Tue&Saks}. Each has a prior probability of 1/4, for being what will apply to SB on a waking day. When she is actually interviewed, she can eliminate one.
And now I suggest that it does not matter, in an interview, what happens differently on H&Tue. SB is interviewed in the context of an interview day, so the question is placed in that context. Claiming it has the context of Sunday Night ignores how the Saks option can only be addressed in a single-day context, and can affect her answer.
I can't tell you whether it is considered to be sound. In my opinion, it is. But I do know that the issues causing the controversy to continue for 23 years are not a part of the actual problem. And so are unnecessary.
If anyone thinks that sounds odd, they should go back and read the question that Elga posed. It does not mention Monday, Tuesday, or that a Tails-only waking follows a mandatory waking. Those were elements he introduced into the problem for his thirder solution.
In the problem as posed, the subject (I'll call her SB, even though in the posed problem it is "you") is woken once, or twice, based on the outcome of a fair coin flip (Heads=once, Tails=twice). Elga enacted that description with a mandatory waking on Monday, and an optional one on Tuesday. This way, could create two partial solutions by revealing two different bits of information that removed one of the three possibilities. The unfortunate side effect of this was that the conditions surrounding Monday and Tuesday are different; thirders require Tuesday to be part of a different outcome, and halfers insist it is the same outcome as Monday+Tails.
And that is what is unnecessary. Instead of that Monday/Tuesday schedule, simply flip two coins (call them C1 and C2) after SB is first put to sleep. Then:
Now turn coin C2 over to show its other face, and repeat these steps.
When SB is woken, she knows that: (A) She is in step 2 of a pass thru these six steps. (B) In step 1, there were four equally-likely states for the two coins: HH, HT, TH, and TT. (C) Since she is awake, the state HH is eliminated but the other three remain equally likely. (D) In only one of those states is C1 showing Heads. So she can confidently state that her credence is 1/3.
The difference between this, and how Elga enacted the problem he posed, is that here there is no ambiguity about how she arrived at her current, awake situation.
The need for distinguishing between SIA and SSA is not needed in the Sleeping Beauty Problem. It was inserted into Adam Elga's problem when he changed it from the one he posed, to the one he solved. I agree that they should have the same answer, which may help in choosing SIA or SSA, but it is not needed. This is what he posed:
"Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?"
The need was created when Elga created a schedule for waking, that treated two days differently. So that "existence" became an alleged issue. There is a better way. First, consider this "little experiment":
The answer should be obvious: From the fact that you are in step 2, as established by being asked a question, the combination HH is eliminated. What happens in step 3 is irrelevant, since the question is not asked there. In only one of the three remaining combinations is coin C1 showing Heads, so there is a 1/3 chance that coin C1 is showing heads.
To implement the experiment Elga proposed - not the one he solved - put SB to sleep on Sunday night, and flip the two coins. On Monday, perform the "little experiment" using the result of the flips. You will need to wake SB if step 2 is executed. What you do in step 3 is still irrelevant, but can include leaving her asleep. Afterwards, if she is awake, put her back to sleep with amnesia. AND THEN TURN COIN C2 OVER. On Tuesday, perform the "little experiment" again, using the modified result of the flips.
SB does not need to consider any other observers than herself to answer the question, because she knows every detail of the "little experiment." If she is awake, and asked a question, the coins were arranged as described in step 1 and she is in step 2. The answer is 1/3.