That intuition pump doesn’t really work for me. The extreme cases seems to make it more apparent that thirding is correct.
There are ethical implications to accepting different assumptions about observation selection effects. For example, SSA has the possible assumption of a near term doomsday. One implication of the SIA, in my view, is that multiple existences is possible.
Super interesting, I have the total opposite intuition and quite strongly. Not sure how I missed these terms in relation to the SB problem (though I think I've heard them used in other, similar contexts). Clearly this is the actual, formalized thing underneath.
For others reading:
Halfer ~ SSA = self-sampling assumption = "All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class."
Thirder ~ SIA = self-indication assumption = "All other things equal, an observer should reason as if they are randomly selected from the set of all possible observers."
If everyone were like you and I the experimenter offered this bet to everyone each time they woke up, I'd make money. The subjects as a group would lose money.
"Waking is not evidence that the coin landed tails." Nobody say it is. It is an opportunity to observe a consequence of the outcomne of a 50:50 random experiment, not an event. The error made by halfers, is confusing the result of the experiment with this opportunity. Since the possibility of this observation depends on the result, it is evidence.
There are two possible observations THAT ARE MADE INDEPENDENT by the amnesia drug. The evidence is that the ability to make an observation is denied in some situations when the result is HEADS, but it can always be made if it is TAILS. So the conditional probabilities of those two results, given that an observation is being made, cannot be the same. "HEADS" has to be diminished relative to "TAILS." This argument does not prove that 1/3 is correct, but it does prove that 1/2 cannot be.
And the real issue is that the setup you cite does not match the actual problem. You can look it up in Adam Elga's paper. It is the way he tried to solve the original problem. The original problem does not mention Monday, or Tuesday, or define a difference between the possibility an observation on those two days. It just says that you (it was phrased on the first person) will be woken once if HEADS and twice if TAILS. Elga first created the scheduling difference, and then eliminated it by "telling" the subject two different pieces of information. As you detailed. There is a better way, that makes each possible observation the same and allows a trivial solution.
On Sunday Night, after SB is asleep, flip two coins. Call them C1 and C2.
On Monday, if both coins are showing HEADS, let SB sleep thru the day. But if either is showing TAILS, wake her, and ask her "What is your credence now for the proposition that coin C1 is showing HEADS?" After she answers, put her back to sleep with amnesia.
On Monday Night, when SB is again (or still) asleep, turn coin C2 over.
On Tuesday Morning, if both coins are showing HEADS, let SB sleep thru the day. But if either is showing TAILS, wake her, and ask her "What is your credence now for the proposition that coin C1 is showing HEADS?" After she answers, put her back to sleep with amnesia.
There are still the two observation opportunities that the original problem asked for. But now they are, as far as probability is concerned, identical. So we can solve that one-day problem, not the two-day one. When the two coins were examined this morning, there were four equally-likely combinations: HH, HT, TH, and TT. But since SB was awakened, she knows that HH is eliminated. The remaining three, HT, TH, and TT are still equally likely, and in only one is coin C1 showing HEADS.
I'm still slightly confused here myself. My initial thoughts were quite strongly in the 1/2 camp, but Arjun makes a compelling case.
Something that confuses me still more- what probability should Beauty ascribe, on waking, to the day being Tuesday? Is it 1/2, 1/3 or 1/4? 1/2 seems clearly wrong to me, but the original proposer of 1/3 seemed to think it correct.
* All experiments feature a Monday awakening. Half of experiments feature a Tuesday one. So Tuesday awakenings are exactly half as common as Monday ones. So the probability must be 1/3.
* There's a 50% chance of Heads, in which case it is Monday for sure. There's a 50% chance of Tails, in which case there's a 50% chance it's Tuesday. So the probability is 1/4.
If you ask me before the experiment, 'What is the probability of the coin landing heads conditional on you waking up as a sleeping beauty test subject,' 1/3. So I would not update throughout the experiment and still be a 'thirder'. If you just asked for the probability of the coin landing heads, I would say 1/2, and then me waking up and being interviewed would be evidence of the type 'I am a sleeping beauty test subject' and would cause me to update to 1/3.
That intuition pump doesn’t really work for me. The extreme cases seems to make it more apparent that thirding is correct.
There are ethical implications to accepting different assumptions about observation selection effects. For example, SSA has the possible assumption of a near term doomsday. One implication of the SIA, in my view, is that multiple existences is possible.
Super interesting, I have the total opposite intuition and quite strongly. Not sure how I missed these terms in relation to the SB problem (though I think I've heard them used in other, similar contexts). Clearly this is the actual, formalized thing underneath.
For others reading:
Halfer ~ SSA = self-sampling assumption = "All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class."
Thirder ~ SIA = self-indication assumption = "All other things equal, an observer should reason as if they are randomly selected from the set of all possible observers."
- from https://en.wikipedia.org/wiki/Anthropic_Bias_(book)#Self-indication_assumption
Might write a brief followup responding to this and a couple other points
If you wake up and I offer you a bet on what day it is, what odds do you accept?
Personally, I try to avoid gambling while under the effects of amnesia drugs. Already lost one house that way, don't need to lose another.
Pretending I was risk-neutral wrt money, 1:1 or better
If everyone were like you and I the experimenter offered this bet to everyone each time they woke up, I'd make money. The subjects as a group would lose money.
For rn im gonna be lazy and just link this twitter thread where I discussed this yesterday: https://twitter.com/RandomSprint/status/1665162050543271936?s=20
but I hope to get out a quick followup addressing this and a few other things
The correct answer is 1/3.
"Waking is not evidence that the coin landed tails." Nobody say it is. It is an opportunity to observe a consequence of the outcomne of a 50:50 random experiment, not an event. The error made by halfers, is confusing the result of the experiment with this opportunity. Since the possibility of this observation depends on the result, it is evidence.
There are two possible observations THAT ARE MADE INDEPENDENT by the amnesia drug. The evidence is that the ability to make an observation is denied in some situations when the result is HEADS, but it can always be made if it is TAILS. So the conditional probabilities of those two results, given that an observation is being made, cannot be the same. "HEADS" has to be diminished relative to "TAILS." This argument does not prove that 1/3 is correct, but it does prove that 1/2 cannot be.
And the real issue is that the setup you cite does not match the actual problem. You can look it up in Adam Elga's paper. It is the way he tried to solve the original problem. The original problem does not mention Monday, or Tuesday, or define a difference between the possibility an observation on those two days. It just says that you (it was phrased on the first person) will be woken once if HEADS and twice if TAILS. Elga first created the scheduling difference, and then eliminated it by "telling" the subject two different pieces of information. As you detailed. There is a better way, that makes each possible observation the same and allows a trivial solution.
On Sunday Night, after SB is asleep, flip two coins. Call them C1 and C2.
On Monday, if both coins are showing HEADS, let SB sleep thru the day. But if either is showing TAILS, wake her, and ask her "What is your credence now for the proposition that coin C1 is showing HEADS?" After she answers, put her back to sleep with amnesia.
On Monday Night, when SB is again (or still) asleep, turn coin C2 over.
On Tuesday Morning, if both coins are showing HEADS, let SB sleep thru the day. But if either is showing TAILS, wake her, and ask her "What is your credence now for the proposition that coin C1 is showing HEADS?" After she answers, put her back to sleep with amnesia.
There are still the two observation opportunities that the original problem asked for. But now they are, as far as probability is concerned, identical. So we can solve that one-day problem, not the two-day one. When the two coins were examined this morning, there were four equally-likely combinations: HH, HT, TH, and TT. But since SB was awakened, she knows that HH is eliminated. The remaining three, HT, TH, and TT are still equally likely, and in only one is coin C1 showing HEADS.
The answer is proven to be 1/3.
I'm still slightly confused here myself. My initial thoughts were quite strongly in the 1/2 camp, but Arjun makes a compelling case.
Something that confuses me still more- what probability should Beauty ascribe, on waking, to the day being Tuesday? Is it 1/2, 1/3 or 1/4? 1/2 seems clearly wrong to me, but the original proposer of 1/3 seemed to think it correct.
* All experiments feature a Monday awakening. Half of experiments feature a Tuesday one. So Tuesday awakenings are exactly half as common as Monday ones. So the probability must be 1/3.
* There's a 50% chance of Heads, in which case it is Monday for sure. There's a 50% chance of Tails, in which case there's a 50% chance it's Tuesday. So the probability is 1/4.
If you ask me before the experiment, 'What is the probability of the coin landing heads conditional on you waking up as a sleeping beauty test subject,' 1/3. So I would not update throughout the experiment and still be a 'thirder'. If you just asked for the probability of the coin landing heads, I would say 1/2, and then me waking up and being interviewed would be evidence of the type 'I am a sleeping beauty test subject' and would cause me to update to 1/3.