81-Days of Failure

The 81-day case gives no reason to abandon SIA

A Redditor who runs the sleeping beauty website has a post claiming to disprove the self-indication assumption. The self-indication assumption is the view of anthropics according to which, given that you exist, you should think there are more total observer moments in existence (observer moments are experiences at a time). Famously, SIA has implications for:

The Sleeping Beauty problem: Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

SIA says you should think the odds are 1/3, because there will be twice as many observer moments if the coin comes up tails. Others disagree. Building on the sleeping beauty problem, the author of the Reddit post provides the following supposed counterexample, which, I shall argue, provides no reason to reject SIA:

The 81-Day Experiment(81D):

There is a building with a circular corridor connected to 81 rooms with identical doors. At the beginning all rooms have blue walls. Then a painter randomly selects an unknown number of 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 thirders, beauty’s own red room is treated differently. As SIA states, finding herself awake is as if she chose a random room from all 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.

I believe the above calculations are straightforward. The same numbers would be obtained as the most likely cases in a bayesian analysis with uniform priors.

However there are contradictions in thirders’ answer. First of all, before opening any doors if beauty is told that R=21 she should expect to see 2 reds if she opens 8 doors. According to thirders after opening 8 doors and actually seeing the expected number of 2 red rooms beauty must estimate R=27 instead of 21. This change of mind cannot be explained.

Imagine that at the beginning of the experiment, beauty knew that were she to sample another room after she was done sampling rooms, it would be red. SIA holds that this is relevantly like the actual situation—she has, from the outset, evidence of an extra red room. One can see that all of OP’s arguments are wrong from the fact that they’d imply that this judgment is contradictory.

If she had this knowledge of an extra red room from the outset and then saw 2 red rooms and 6 blue ones, she should estimate R=27. Yet R=21 best explains the rooms she sees! How can this work? One could similarly claim “This change of mind cannot be explained.” But there’s no change of mind because, before the sampling, she starts with a higher prior in views according to which there are more red rooms. Upon 6 blue and 2 red rooms, R goes down, because it’s fewer red than she expected at the outset to observe if she opened 8 rooms.

It’s true that the theory that R=21 better explains the doors she samples. But SIAers claim that this is not the only evidence she has. Beauty also has the evidence that she is awake now. Thus, thirders happily accept that R=21 better explains part of the evidence, but reject think there’s extra evidence that must be considered.

Secondly, estimating R=27 means if beauty opens another 8 random doors she should expect to see 24/72×8=2.67 red rooms. This means after beauty saw 2 reds in the first 8 random rooms she would expect to see about 3 reds if she choose another 8 rooms. I fail to see how can this be justified.

Once again, this involves erroneously assuming that she has no other evidence. It’s true that if her only evidence were the 8 rooms, it would be irrational to predict the next sample would have more reds. But if she has other evidence—from her present existence—then she should think that the sample undercounted the total share that’s red. This is what happens when one only accounts for part of the evidence.

Last but not least, because beauty believes the 9 rooms beauty knows is a fair sample of all 81 rooms, 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 average. This supernatural predicting power is a conclusive evidence against SIA and thirders’ argument.

Once again, this involves erroneously assuming she has no other evidence. If one has evidence that points in one direction, then they have reason to expect some sample to be less than what they’ll ultimately conclude about the evidence. For example, suppose that there’s a jar with some number of red and blue jelly beans, such that it contains both. I’ve already drawn 3 jellybeans from the jar and they were all red. Now, if I draw 100 more, I have reason to expect the sample to undercount the number of red jellybeans—at least, relative to what I should think about the sample. If I start out, prior to observing some evidence, with some evidence favoring theory A over B, then it’s expected that after I gather the evidence, the evidence alone will be biased in favor of B relative to the total evidence. No psychic powers are involved, just reasoning in exactly the way one would if they had evidence for the result of some room prior to looking at the other rooms. Which, SIAers claim, beauty does.