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 rooms. At this point in time, the selector’s state of knowledge about the rooms, regardless of what you consider prior and what posterior, is in the same position as beauty’s after she wakes up. (from the thirder perspective, which I generally agree with in this case). Then they both sample 8 more rooms. The selector considers this an unbiased sample of the remaining 80 rooms. After both have taken this additional sample of 8, they again agree. Since they still agree, beauty must also consider the 8 rooms to be an unbiased sample of the remaining 80 rooms. Beauty’s reasoning and the selector’s are the same regarding the additional 8 rooms, and Beauty has no more “supernatural predicting power” than the selector.
About only thirding getting the attention: my apologies for contributing to this asymetry. For me, the issue is, I found the perspectivism posts at least initially hard to understand, and since subjectively I feel I already know the correct way to handle this sort of problem, that reduces my motivation to persevere and figure out what you are saying. I’ll try to get around to carefully reading them and providing some response eventually (no time right now).
Ok, I should have use my words more carefully. We meant the same thing. When I say beauty think the 8 rooms are unbiased sample I meant what I listed as C: It is an unbiased for the other 80 rooms. So yes to what you said, sorry for the confusion. it is obvious because it is a simple random sample chosen from the 80 rooms. So that part there is no disagreement. The disagreement between the two is about whether or not the 9 rooms are an unbiased sample. Beauty as a thirder should not think it is unbiased but bases her estimation on it anyway to answer the question from the selector’s perspective. If she does not answer from selector’s perspective she would use the 8 rooms to estimate the reds in the other 80 rooms and then add her own room in, as halfers does.
Regarding the selector chooses a room and finds out it is red. Again they agree on whether or not the 8 rooms are unbiased, however because the first room is always red for beauty but not so for the selector they see the 9 rooms differently. From beauty’s perspective dividing the 9 rooms into 2 parts and she gets a unbiased sample (8 rooms) and a red room. It is not so for the selector. We can list the three points from the selector’s perspective and it poses no problem at all.
A: the 9 room is an unbiased sample for 81 rooms
B: the first room is randomly selected from all rooms
C: the other 8 rooms is an unbiased sample for other 80 rooms.
alternatively we can divid the 9 rooms as follows:
A: the 9 rooms is an unbiased sample for 81 rooms
B: the first red room he saw (if he saw one) is always red
C: the other 8 rooms in the sample is biased towards blue
Either way there is no problem. In terms of the predicting power, think of it this way. Once the selector sees a red room he knows if he ignore it and only consider the other 8 rooms then the sample is biased towards blue, nothing supernatural. However, for beauty if she thinks the 9 rooms are unbiased then the 8 rooms she chooses must be biased even though they are selected at random. Hence the “supernatural”. It is meant to point out for beauty the 9 and 8 rooms cannot be unbiased at same time. Since you already acknowledged the 9 rooms is biased (for her perspective at least), then yes she does not have supernatural predicting power of course.
I guess the bottomline is because they acquire their information differently, the selector and thirder beauty must disagree somewhere. Either on the numerical value of estimate, or on if a sample is biased.
About the perspectivism posts. The concept is actually quite simple: each beauty only counts what she experienced/remembered. But I feel maybe I’m not doing a good job explaining it. Anyway, thank you for promising to check it out.
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.
Thanks for the kind words.
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 rooms. At this point in time, the selector’s state of knowledge about the rooms, regardless of what you consider prior and what posterior, is in the same position as beauty’s after she wakes up. (from the thirder perspective, which I generally agree with in this case). Then they both sample 8 more rooms. The selector considers this an unbiased sample of the remaining 80 rooms. After both have taken this additional sample of 8, they again agree. Since they still agree, beauty must also consider the 8 rooms to be an unbiased sample of the remaining 80 rooms. Beauty’s reasoning and the selector’s are the same regarding the additional 8 rooms, and Beauty has no more “supernatural predicting power” than the selector.
About only thirding getting the attention: my apologies for contributing to this asymetry. For me, the issue is, I found the perspectivism posts at least initially hard to understand, and since subjectively I feel I already know the correct way to handle this sort of problem, that reduces my motivation to persevere and figure out what you are saying. I’ll try to get around to carefully reading them and providing some response eventually (no time right now).
Ok, I should have use my words more carefully. We meant the same thing. When I say beauty think the 8 rooms are unbiased sample I meant what I listed as C: It is an unbiased for the other 80 rooms. So yes to what you said, sorry for the confusion. it is obvious because it is a simple random sample chosen from the 80 rooms. So that part there is no disagreement. The disagreement between the two is about whether or not the 9 rooms are an unbiased sample. Beauty as a thirder should not think it is unbiased but bases her estimation on it anyway to answer the question from the selector’s perspective. If she does not answer from selector’s perspective she would use the 8 rooms to estimate the reds in the other 80 rooms and then add her own room in, as halfers does.
Regarding the selector chooses a room and finds out it is red. Again they agree on whether or not the 8 rooms are unbiased, however because the first room is always red for beauty but not so for the selector they see the 9 rooms differently. From beauty’s perspective dividing the 9 rooms into 2 parts and she gets a unbiased sample (8 rooms) and a red room. It is not so for the selector. We can list the three points from the selector’s perspective and it poses no problem at all.
A: the 9 room is an unbiased sample for 81 rooms
B: the first room is randomly selected from all rooms
C: the other 8 rooms is an unbiased sample for other 80 rooms.
alternatively we can divid the 9 rooms as follows:
A: the 9 rooms is an unbiased sample for 81 rooms
B: the first red room he saw (if he saw one) is always red
C: the other 8 rooms in the sample is biased towards blue
Either way there is no problem. In terms of the predicting power, think of it this way. Once the selector sees a red room he knows if he ignore it and only consider the other 8 rooms then the sample is biased towards blue, nothing supernatural. However, for beauty if she thinks the 9 rooms are unbiased then the 8 rooms she chooses must be biased even though they are selected at random. Hence the “supernatural”. It is meant to point out for beauty the 9 and 8 rooms cannot be unbiased at same time. Since you already acknowledged the 9 rooms is biased (for her perspective at least), then yes she does not have supernatural predicting power of course.
I guess the bottomline is because they acquire their information differently, the selector and thirder beauty must disagree somewhere. Either on the numerical value of estimate, or on if a sample is biased.
About the perspectivism posts. The concept is actually quite simple: each beauty only counts what she experienced/remembered. But I feel maybe I’m not doing a good job explaining it. Anyway, thank you for promising to check it out.