This is the second part of my argument. It mainly involves a counter example to SIA and Thirdism.
Different part of the argument can be found here: I, II, III, IV.
The 81-Day Experiment(81D):
There is a circular corridor connected to 81 rooms with identical doors. At the beginning all rooms have blue walls. A random number R is generated between 1 and 81. Then a painter randomly selects R rooms and paint them red. Beauty would be put into a drug induced sleep lasting 81 day, spending one day in each room. An experimenter would wake her up if the room she currently sleeps in is red and let her sleep through the day if the room is blue. Her memory of each awakening would be wiped at the end of the day. Each time after beauty wakes up she is allowed to exit her room and open some other doors in the corridor to check the colour of those rooms. Now suppose one day after opening 8 random doors she sees 2 red rooms and 6 blue rooms. How should beauty estimate the total number of red rooms(R).
For halfers, waking up in a red room does not give beauty any more information except that R>0. Randomly opening 8 doors means she took a simple random sample of size 8 from a population of 80. In the sample 2 rooms (1/4) are red. Therefore the total number of red rooms(R) can be easily estimated as 1/4 of the 80 rooms plus her own room, 21 in total.
For thirders, beauty's own room is treated differently.As SIA states, finding herself awake is as if she chose a random room from the 81 rooms and find out it is red. Therefore her room and the other 8 rooms she checked are all in the same sample. This means she has a simple random sample of size 9 from a population of 81. 3 out of 9 rooms in the sample (1/3) are red. The total number of red rooms can be easily estimated as a third of the 81 rooms, 27 in total.
If a bayesian analysis is performed R=21 and R=27 would also be the case with highest credence according to halfers and thirders respectively. It is worth mentioning if an outside Selector randomly chooses 9 rooms and check them, and it just so happens those 9 are the same 9 rooms beauty saw (her own room plus the 8 randomly chosen rooms), the Selector would estimate R=27 and has the highest credence for R=27. Because he and the beauty has the exact same information about the rooms their answer would not change even if they are allowed to communicate. So again, there will be a perspective disagreement according to halfers but not according to thirders. Same as mentioned in part I.
However, thirder's estimation is very problematic. Because beauty believes the 9 rooms she knows is a fair sample of all 81 rooms (since she used it in statistical estimation), it means red rooms (and blue rooms) are not systematically over- or under-represented. Since beauty is always going to wake up in a red room, she has to conclude the other 8 rooms is not a fair sample. Red rooms have to be systematically underrepresent in those 8 rooms. This means even before beauty decides which doors she wants to open we can already predict with certain confidence that those 8 rooms is going to contains less reds than the average of the 80 suggests. This supernatural predicting power is a strong evidence against SIA and thirding.
The argument can also be structured this way. Consider the following three statements:
A: The 9 rooms is an unbiased sample of the 81 rooms.
B: Beauty is guaranteed to wake up in a red room
C: The 8 rooms beauty choose is an unbiased sample of the other 80 rooms.
These statements cannot be all true at the same time. Thirders accept A and B meaning they must reject C. In fact they must conclude the 8 rooms she choose would be biased towards blue. This contradicts the fact that the 8 rooms are randomly chosen.
(EDIT Aug 1. I think the best answer thirders shall give is accept C since it is obviously a simple random sample. Adding another red room to this will make the 9 rooms biased from his perspective. However they can argue if a selector saw the same 9 rooms through random selection then it is unbiased from the selector's perspective. Thirders could argue she must answer from the selector's perspective instead of her own. Main reason being she is undergoing potential memory wipes so her perspective is somewhat "compromised". However, with this explanation thirder must confirm the perspective disagreement between beauty and the selector about whether or not the 9 rooms are biased. It also utilize perspective reasoning followed. In another word perspective disagreement is not unique to halfers and shall not be treated as a weakness.)
It is also easy to see why beauty should not estimate R the same way as the selector does. There are about 260 billion distinct combinations to pick 9 rooms out of 81. The selector has a equal chance to see any one of those 260 billion combinations. Beauty on the other hand could only possibility see a subset of the combinations. If a combination does not contains a red room, beauty would never see it. Furthermore, the more red rooms a combination contains the more awakening it has leading to a greater chance for a beauty to select the said combination. Therefore while the same 9 rooms is a unbiased sample for the selector it is a sample biased towards red for beauty.
(EDIT Aug 1. We can show this another way. Let the selector, halfer beauty and thirder beauty do a large number repeated estimation on the same set of rooms. The selector and halfer's estimations would be concentrated around the true value of R, where as thirders answer would be concentrated on some value larger.)
One might want to argue after the selector learns a beauty has the knowledge of the same 9 rooms he should lower his estimation of R to the same as beauty’s. After all beauty could only know combinations in a subset biased towards red. The selector should also reason his sample is biased towards red. This argument is especially tempting for SSA supporters since if true it means their answer also yields no disagreements. Sadly this notion is wrong, the selector ought to remain his initial estimation. To the selector a beauty knowing the same 9 rooms simply means after waking up in one of the red rooms in his sample, beauty made a particular set of random choices coinciding said sample. It offers him no new information about the other rooms. This point can be made clearer if we look at how people reach to an agreement in an ordinary problem. Which would be shown by another thought experiment in the next part.
Very clear argument and many good points. Appreciate the effort.
Regarding your position on thirders vs halfers, I think it is a completely reasonable position and I agree with the analysis about when halfers are correct and when thirders are correct. However to me it seems to treat Sleeping Beauty more as a decision making problem rather than a probability problem. Maybe one's credence without relating consequences is not defined. However that seems counter intuitive to me. Naturally one should have a belief about the situation and her decisions should depend on it as well as her objective (how much beauty cases about other copies) and the payoff structure (is the money reward depends only on her own answer, or all correct answers or accuracy rate etc). If that's the case, there should exist a unique correct answer to the problem.
About how should beauty estimate R and treat the samples, I would say that's the best position for a thirder to take. In fact that's the same position I would take too. If I may reword it slightly, see if you agrees with this version: The 8 rooms is a unbiased sample for beauty, that is too obvious to argue otherwise. Her own room is always red so the 9 rooms is obviously biased for her. However from (an imaginary) selector's perspective if he finds the same 9 rooms it is an unbiased sample. Thirders think she should answer from the selector's perspective, (I think the most likely reason being she is repeatedly memory wiped makes her perspective somewhat "compromised") therefore she would estimate R to be 27. Is this version something you would agree?
In this version I highlighted the disagreement between the selector and beauty, the disagreement is not some numerical value but they disagree on whether a sample is biased. In my 4 posts all I'm trying to do is arguing for the validity and importance of perspective disagreement. If we recognize the existence of this disagreement and let each agent answers from her own perspective we get another system of reasoning different from SIA or SSA. It provides an argument for double halving, give a framework where frequentist and bayesians agrees with each other, reject Doomsday Argument, disagree with Presumptuous Philosopher, and rejects the Simulation Argument. I genuinely think this is the explanation to sleeping beauty problem as well as many problems related to anthropic reasoning. Sadly only the part arguing against thirding gets some attention.
Anyways, I digressed. Bottomline is, though I do no think it is the best position, I feel your argument is reasonable and well thought. I can understand it if people want to take it as their position.
However, I don't agree. The additional 8 rooms is an unbiased sample of the remaining 80 rooms for beauty. The additional 8 rooms is only an unbiased sample of the full set of 81 rooms for beauty if the first room is also an unbiased sample (but I would not consider it a sample but part of the prior).
Actually I found a better argument against your original anti-thirder argument, regardless of where the prior/posterior line is drawn:
Imagine that the selector happened to encounter a red room first, before checking out the other 8 ... (read more)
This is the second part of my argument. It mainly involves a counter example to SIA and Thirdism.
Different part of the argument can be found here: I, II, III, IV.
The 81-Day Experiment(81D):
There is a circular corridor connected to 81 rooms with identical doors. At the beginning all rooms have blue walls. A random number R is generated between 1 and 81. Then a painter randomly selects R rooms and paint them red. Beauty would be put into a drug induced sleep lasting 81 day, spending one day in each room. An experimenter would wake her up if the room she currently sleeps in is red and let her sleep through the day if the room is blue. Her memory of each awakening would be wiped at the end of the day. Each time after beauty wakes up she is allowed to exit her room and open some other doors in the corridor to check the colour of those rooms. Now suppose one day after opening 8 random doors she sees 2 red rooms and 6 blue rooms. How should beauty estimate the total number of red rooms(R).
For halfers, waking up in a red room does not give beauty any more information except that R>0. Randomly opening 8 doors means she took a simple random sample of size 8 from a population of 80. In the sample 2 rooms (1/4) are red. Therefore the total number of red rooms(R) can be easily estimated as 1/4 of the 80 rooms plus her own room, 21 in total.
For thirders, beauty's own room is treated differently.As SIA states, finding herself awake is as if she chose a random room from the 81 rooms and find out it is red. Therefore her room and the other 8 rooms she checked are all in the same sample. This means she has a simple random sample of size 9 from a population of 81. 3 out of 9 rooms in the sample (1/3) are red. The total number of red rooms can be easily estimated as a third of the 81 rooms, 27 in total.
If a bayesian analysis is performed R=21 and R=27 would also be the case with highest credence according to halfers and thirders respectively. It is worth mentioning if an outside Selector randomly chooses 9 rooms and check them, and it just so happens those 9 are the same 9 rooms beauty saw (her own room plus the 8 randomly chosen rooms), the Selector would estimate R=27 and has the highest credence for R=27. Because he and the beauty has the exact same information about the rooms their answer would not change even if they are allowed to communicate. So again, there will be a perspective disagreement according to halfers but not according to thirders. Same as mentioned in part I.
However, thirder's estimation is very problematic. Because beauty believes the 9 rooms she knows is a fair sample of all 81 rooms (since she used it in statistical estimation), it means red rooms (and blue rooms) are not systematically over- or under-represented. Since beauty is always going to wake up in a red room, she has to conclude the other 8 rooms is not a fair sample. Red rooms have to be systematically underrepresent in those 8 rooms. This means even before beauty decides which doors she wants to open we can already predict with certain confidence that those 8 rooms is going to contains less reds than the average of the 80 suggests. This supernatural predicting power is a strong evidence against SIA and thirding.
The argument can also be structured this way. Consider the following three statements:
A: The 9 rooms is an unbiased sample of the 81 rooms.
B: Beauty is guaranteed to wake up in a red room
C: The 8 rooms beauty choose is an unbiased sample of the other 80 rooms.
These statements cannot be all true at the same time. Thirders accept A and B meaning they must reject C. In fact they must conclude the 8 rooms she choose would be biased towards blue. This contradicts the fact that the 8 rooms are randomly chosen.
(EDIT Aug 1. I think the best answer thirders shall give is accept C since it is obviously a simple random sample. Adding another red room to this will make the 9 rooms biased from his perspective. However they can argue if a selector saw the same 9 rooms through random selection then it is unbiased from the selector's perspective. Thirders could argue she must answer from the selector's perspective instead of her own. Main reason being she is undergoing potential memory wipes so her perspective is somewhat "compromised". However, with this explanation thirder must confirm the perspective disagreement between beauty and the selector about whether or not the 9 rooms are biased. It also utilize perspective reasoning followed. In another word perspective disagreement is not unique to halfers and shall not be treated as a weakness.)
It is also easy to see why beauty should not estimate R the same way as the selector does. There are about 260 billion distinct combinations to pick 9 rooms out of 81. The selector has a equal chance to see any one of those 260 billion combinations. Beauty on the other hand could only possibility see a subset of the combinations. If a combination does not contains a red room, beauty would never see it. Furthermore, the more red rooms a combination contains the more awakening it has leading to a greater chance for a beauty to select the said combination. Therefore while the same 9 rooms is a unbiased sample for the selector it is a sample biased towards red for beauty.
(EDIT Aug 1. We can show this another way. Let the selector, halfer beauty and thirder beauty do a large number repeated estimation on the same set of rooms. The selector and halfer's estimations would be concentrated around the true value of R, where as thirders answer would be concentrated on some value larger.)
One might want to argue after the selector learns a beauty has the knowledge of the same 9 rooms he should lower his estimation of R to the same as beauty’s. After all beauty could only know combinations in a subset biased towards red. The selector should also reason his sample is biased towards red. This argument is especially tempting for SSA supporters since if true it means their answer also yields no disagreements. Sadly this notion is wrong, the selector ought to remain his initial estimation. To the selector a beauty knowing the same 9 rooms simply means after waking up in one of the red rooms in his sample, beauty made a particular set of random choices coinciding said sample. It offers him no new information about the other rooms. This point can be made clearer if we look at how people reach to an agreement in an ordinary problem. Which would be shown by another thought experiment in the next part.
Part III can be found at here.