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.
Very clear argument, thank you for the reply.
The question is if we do not use bayesian reasoning, just use statistics analysis can we still get an unbiased estimation? The answer is of course yes. Using fair sample to estimate population is as standard as it gets. The main argument is of course what is the fair sample. Depending on the answer we get estimation of r=21 or 27 respectively.
SIA states we should treat beauty's own room as a randomly selected from all rooms. By applying this idea in bayesian analysis is how we get thirdism. To oversimplify it: we shall reason as some selector randomly chose a day and find beauty awake, which in itself is a coincidence. However there is no reason for SIA to apply only to bayesian analysis but not statistical analysis. If we use SIA reasoning in statistical analysis, treating her own room as randomly selected from all 81 rooms, then the 9 rooms are all part of a simple random sample, which by definition is unbiased. There is no baye's rule or conditioning involved because here we are not treating it as a probability problem. Beauty's own red room is just a coincidence as in bayesian analysis, it suggest a larger number of reds the same way the other 2 red rooms does.
If one want to argue those 9 rooms are biased, why not use the same logic in a bayesian analysis? Borrowing cousin_it's example. If there are 3 rooms with the number of red rooms uniformly distributed between 1 and 3. If beauty wakes up and open another door and sees another red what should her credence of R=3 be? If I'm not mistaken thirders will say 3/4. Because by randomly selecting 2 room out of 3 and both being red there are 3 ways for R=3 and 1 way for R=2. Here thirders are treating her own room the same way as the second room. And the two rooms are thought to be randomly selected aka unbiased. If one argues the 2 rooms are biased towards red because her own room is red, then the calculation above is no longer valid.
Even if one takes the unlikely position that SIA is only applicable in bayesian but not statistical analysis there are still strange consequences. I might be mistaken but in problems of simple sampling, in general, not considering some round off errors, the statistical estimation would also be the case with highest probability in a bayesian analysis with an uniform prior. By using SIA in a bayesian analysis, we get R=27 as the most likely case. However statistics gives an estimate of R=21. This difference cannot be easily explained.
To answer the last part of your statement. If beauty randomly opens 8 doors and found them all red then she has a sample of pure red. By simple statistics she should give R=81 as the estimation. Halfer and thirders would both agree on that. If they do a bayesian analysis R=81 would also be the case with the highest probability. I'm not sure where 75 comes from I'm assuming by summing the multiples of probability and Rs in the bayesian analysis? But that value does not correspond to the estimation in statistics. Imagine you randomly draw 20 beans from a bag and they are all red, using statistics obviously you are not going to estimate the bag contains 90% red bean.
Sorry for the slow reply.
The 8 rooms are definitely the unbiased sample (of your rooms with one red room subtracted).
I think you are making two mistakes:
First, I think you're too focused on the nice properties of an unbiased sample. You can take an unbiased sample all you want, but if we know information in addition to the sample, our best estimate might not be the average of the sample! Suppose we have two urns, urn A has 10 red balls and 10 blue balls, while urn B has 5 red balls and 15 blue balls. We choose an urn by rolling a die, such that we have a 5... (read more)