Thirder and the selector have the exact same prior and posteriors. Their bayesian analysis are exactly the same.
Think from the selector’s perspective. He randomly opens 9 out of the 81 rooms and found 3 red. Say he decided to perform a bayesian analysis. As stated in the question he starts from an uniform prior and updates it with the 9 rooms as new information. I will skip the calculation but in the end he concluded R=27 has the highest probability. Now think from the thirder’s perspective. As SIA states she is treating her own room as randomly selected from all the rooms. Therefore she is treating all of the 9 rooms as randomly selected from the 81 rooms. The new information she has is exactly the same as the selector. Starting from an uniform prior and updates on those new information she would get the same pdf as the selector with R=27 has the highest probability. The two of them must agree just as in the original sleeping beauty problem.
Now suppose instead of doing a bayesian analysis, the selector just want to perform statistical analysis. He wants to get a fair estimate of R. It is clear he has a simple random sample with sample size 9 and 3 of which are red. He can estimate the R of the population as 3/9x81=27. It is unsurprising he got the same number as his bayesian analysis since he started from an uniform prior. Until now, all good.
The problem, however, starts when the thirder wants to use his sample in simple statistics. If he uses SIA reasoning as he did in bayesian analysis he would treat the 9 rooms as a simple random sample the same way as the selector did and give R=27 as the unbiased estimation. By doing so he accepts the 9 rooms as an unbiased sample which leads to the problems I discussed in the main post.
For the priors,. I would consider Beauty’s expectations from the problem definition before she takes a look at anything to be a prior, i.e. she expects 81 times higher probability of R=81 than R=1 right from the start.
SIA states that you should expect to be randomly selected from the set of possible observers. That doesn’t imply that you are in a postion randomly selected from some other set. (only if observers are randomly selected from that set). Here, observers start in red rooms only, so clearly, you can’t expect your room to be randomly selected colour if you believe in SIA.
For the priors,. I would consider Beauty’s expectations from the problem definition before she takes a look at anything to be a prior, i.e. she expects 81 times higher probability of R=81 than R=1 right from the start.
In the original sleeping beauty problem, what is the prior for H according to a thirder? It must be 1⁄2. In fact saying she expects 2 times higher probability of T than H right from the start means she should conclude P(H)=1/3 before going to sleep on Sunday. That is used as a counter argument by halfers. Thirders are arguing after waking up in the experiment, beauty should update her probability as waking up is new information. T being 2 times more likely than H is a posterior.
If you think thirders should reject A based on your interpretation of SIA, then what is a fair estimation of R according to thirders? Should they use a biased sample of 9 rooms and estimate 27, or estimate 21 and disagree with the selector having the same information?
Well argued, you’ve convinced me that most people would probably define what’s prior and what’s posterior the way you say. Nonetheless, I don’t agree that what’s prior and what’s posterior should be defined the way you say. I see this sort of info as better thought of as a prior (precisely because waking up shouldn’t be thought of as new info) [edit: clarification below]. I don’t regard the mere fact that the brain instantiating the mind having this info is physically continuous with an earlier-in-time brain instantiating a mind with different info as sufficient to not make it better thought of as a prior.
Some clarification on my actual beliefs here: I’m not a conventional thirder believing in the conventional SIA. I prefer, let’s call it, “instrumental epistemic rationality”. I weight observers, not necessarily equally, but according to how much I care about the accuracy of the relevant beliefs of that potential observer. If I care equally about the beliefs of the different potential observers, then this reduces to SIA. But there are many circumstances where one would not care equally, e.g. one is in a simulation and another is not, or one is a Boltzmann brain and another is not.
Now, I generally think that thirdism is correct, because I think that, given the problem definition, for most purposes it’s more reasonable to value the correctness of the observers equally in a sleeping beauty type problem. E.g. if Omega is going to bet with each observer, and beauty’s future self collects the sum of the earnings of both observers in the case there are two of them, then 1⁄3 is correct. But if e.g. the first instance of the two observer case is valued at zero, or if for some bizarre reason you care equally about the average of the correctness of the observers in each universe regardless of differences in numbers, then 1⁄2 is correct.
Now, I’ll deal with your last paragraph from my perspective, The first room isn’t a sample, it’s guaranteed red. If you do regard it as a sample, it’s biased in the red direction (maximally) and so should have zero weight. The prior is that the probability of R is proportional to R. The other 8 rooms are an unbiased sample of the remaining rooms. The set of 9 rooms is a biased sample (biased in the red direction) such that it provides the same information as the set of 8 rooms. So use the red-biased prior and the unbiased (out of the remaining rooms after the first room is removed) 8 room sample to get the posterior esimate. This will result in the same answer the selector gets, because you can imagine the selector found a red room first and then break down the selector’s information into that first sample and a second unbiased sample of 8 of the remaining rooms.
Edit: I didn’t explain my concept of prior v. posterior clearly. To me, it’s conceptual not time-based in nature. For a set problem like this, what someone knows from the problem definition, from the point of view of their position in the problem, is the prior. What they then observe leads to the posterior. Here, waking sleeping beauty learns nothing on waking up that she does not know from the problem definition, given that she is waking up in the problem. So her beliefs at this point are the prior. Of course, her beliefs are different from sleeping beauty before she went to sleep, due to the new info. That new info told her she is within the problem, when she wasn’t before, so she updated her beliefs to new beliefs which would be a posterior belief outside the context of the problem, but within the context of the problem constitute her prior.
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.
Thirder and the selector have the exact same prior and posteriors. Their bayesian analysis are exactly the same.
Think from the selector’s perspective. He randomly opens 9 out of the 81 rooms and found 3 red. Say he decided to perform a bayesian analysis. As stated in the question he starts from an uniform prior and updates it with the 9 rooms as new information. I will skip the calculation but in the end he concluded R=27 has the highest probability. Now think from the thirder’s perspective. As SIA states she is treating her own room as randomly selected from all the rooms. Therefore she is treating all of the 9 rooms as randomly selected from the 81 rooms. The new information she has is exactly the same as the selector. Starting from an uniform prior and updates on those new information she would get the same pdf as the selector with R=27 has the highest probability. The two of them must agree just as in the original sleeping beauty problem.
Now suppose instead of doing a bayesian analysis, the selector just want to perform statistical analysis. He wants to get a fair estimate of R. It is clear he has a simple random sample with sample size 9 and 3 of which are red. He can estimate the R of the population as 3/9x81=27. It is unsurprising he got the same number as his bayesian analysis since he started from an uniform prior. Until now, all good.
The problem, however, starts when the thirder wants to use his sample in simple statistics. If he uses SIA reasoning as he did in bayesian analysis he would treat the 9 rooms as a simple random sample the same way as the selector did and give R=27 as the unbiased estimation. By doing so he accepts the 9 rooms as an unbiased sample which leads to the problems I discussed in the main post.
We may just be arguing over definintions.
For the priors,. I would consider Beauty’s expectations from the problem definition before she takes a look at anything to be a prior, i.e. she expects 81 times higher probability of R=81 than R=1 right from the start.
SIA states that you should expect to be randomly selected from the set of possible observers. That doesn’t imply that you are in a postion randomly selected from some other set. (only if observers are randomly selected from that set). Here, observers start in red rooms only, so clearly, you can’t expect your room to be randomly selected colour if you believe in SIA.
In the original sleeping beauty problem, what is the prior for H according to a thirder? It must be 1⁄2. In fact saying she expects 2 times higher probability of T than H right from the start means she should conclude P(H)=1/3 before going to sleep on Sunday. That is used as a counter argument by halfers. Thirders are arguing after waking up in the experiment, beauty should update her probability as waking up is new information. T being 2 times more likely than H is a posterior.
If you think thirders should reject A based on your interpretation of SIA, then what is a fair estimation of R according to thirders? Should they use a biased sample of 9 rooms and estimate 27, or estimate 21 and disagree with the selector having the same information?
Well argued, you’ve convinced me that most people would probably define what’s prior and what’s posterior the way you say. Nonetheless, I don’t agree that what’s prior and what’s posterior should be defined the way you say. I see this sort of info as better thought of as a prior (precisely because waking up shouldn’t be thought of as new info) [edit: clarification below]. I don’t regard the mere fact that the brain instantiating the mind having this info is physically continuous with an earlier-in-time brain instantiating a mind with different info as sufficient to not make it better thought of as a prior.
Some clarification on my actual beliefs here: I’m not a conventional thirder believing in the conventional SIA. I prefer, let’s call it, “instrumental epistemic rationality”. I weight observers, not necessarily equally, but according to how much I care about the accuracy of the relevant beliefs of that potential observer. If I care equally about the beliefs of the different potential observers, then this reduces to SIA. But there are many circumstances where one would not care equally, e.g. one is in a simulation and another is not, or one is a Boltzmann brain and another is not.
Now, I generally think that thirdism is correct, because I think that, given the problem definition, for most purposes it’s more reasonable to value the correctness of the observers equally in a sleeping beauty type problem. E.g. if Omega is going to bet with each observer, and beauty’s future self collects the sum of the earnings of both observers in the case there are two of them, then 1⁄3 is correct. But if e.g. the first instance of the two observer case is valued at zero, or if for some bizarre reason you care equally about the average of the correctness of the observers in each universe regardless of differences in numbers, then 1⁄2 is correct.
Now, I’ll deal with your last paragraph from my perspective, The first room isn’t a sample, it’s guaranteed red. If you do regard it as a sample, it’s biased in the red direction (maximally) and so should have zero weight. The prior is that the probability of R is proportional to R. The other 8 rooms are an unbiased sample of the remaining rooms. The set of 9 rooms is a biased sample (biased in the red direction) such that it provides the same information as the set of 8 rooms. So use the red-biased prior and the unbiased (out of the remaining rooms after the first room is removed) 8 room sample to get the posterior esimate. This will result in the same answer the selector gets, because you can imagine the selector found a red room first and then break down the selector’s information into that first sample and a second unbiased sample of 8 of the remaining rooms.
Edit: I didn’t explain my concept of prior v. posterior clearly. To me, it’s conceptual not time-based in nature. For a set problem like this, what someone knows from the problem definition, from the point of view of their position in the problem, is the prior. What they then observe leads to the posterior. Here, waking sleeping beauty learns nothing on waking up that she does not know from the problem definition, given that she is waking up in the problem. So her beliefs at this point are the prior. Of course, her beliefs are different from sleeping beauty before she went to sleep, due to the new info. That new info told her she is within the problem, when she wasn’t before, so she updated her beliefs to new beliefs which would be a posterior belief outside the context of the problem, but within the context of the problem constitute her prior.
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.