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The “Sleeping Beauty Problem” has obviously attracted mixed responses – some people say 1/3 and others say 1/2. I’m having trouble working out which group I agree with. In the meantime, here is another thought experiment designed to help us decide if we trust the SIA or the main alternative, the Self-Sampling Assumption (SSA), suggested by Nick Bostrum.
“II. POSSIBLE VS. EXISTING
A. God’s Coin Toss
The crux of the matter can be described by a “God’s Coin Toss” experiment [3,6]. Suppose that God tosses a fair coin. If it comes up heads, he creates ten people, each in their own room. If tails, he creates one thousand people, each in their own room. The rooms are numbered 1-10 or 1-1000. The people cannot see or communicate with the other rooms. Suppose that you know all this, and you discover that you are in one of the first ten rooms. How should you reason that the coin fell? Leslie and Bostrom argue as follows. Before you look at your room number, you should think that since the coin was fair the chance of heads was 1/2. Now if the coin was heads, then of course you would be in one of the first ten rooms. However, if the coin was tails, the chance to be in one of the first ten rooms is 1/100. Thus, according to Eq. (2), you should now believe that the coin was heads with probability 0.99. The alternative argument runs as follows. Before you look at your room number, you should think that the probability of heads is 0.99. There are one thousand possible people who would be right with that belief, whereas only ten would be right with the belief in heads. When you look at your room number, you should then update your probabilities using Eq. (2). The result is that in the end you think the chance is 1/2 that the coin was heads. Another way to say the same thing is that there are ten ways to have the coin heads and you in a room in the first ten, and ten ways to have the coin tails and you in a room in the first ten, and thus the chances for heads and tails are equal. The difference here hinges on whether one considers possible people in the same ways that one considers actual people. If instead of flipping a coin, God creates both sets of rooms, then Leslie and Bostrom and I all agree that you should think it much more probable that you are in the large set before you look at your room number, and equally probable afterward. Treating the two possibilities in the same way as two sets of actual observers implies the Self-Indication Assumption: the existence of a large number of observers in a possible universe increases the chance to find oneself in that universe.
I will argue below that the equal treatment of possible and actual observers is correct.
Here’s the craziest and most fascinating thing I’ve learnt in ages. Strap yourselves in:
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?
First answer: 1/2, of course! Initially you were certain that the coin was fair, and so initially your credence in the coin’s landing Heads was 1/2. Upon being awakened, you receive no new information (you knew all along that you would be awakened). So your credence in the coin’s landing Heads ought to remain 1/2.
Second answer: 1/3, of course! Imagine the experiment repeated many times. Then in the long run, about 1/3 of the wakings would be Heads-wakings — wakings that happen on trials in which the coin lands Heads. So on any particular waking, you should have credence 1/3 that that waking is a Heads-waking, and hence have credence 1/3 in the coin’s landing Heads on that trial. This consideration remains in force in the present circumstance, in which the experiment is performed just once.
I will argue that the correct answer is 1/3.
Do you agree that the correct answer is 1/3? If so, you have just used the Self-Indication Assumption, which says “Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.” This seems very intuitive in the case above, but it can be taken to less obvious extremes.
For example, let’s imagine we have two remaining theories about the nature of the universe. One says that the universe is big: it contains a trillion galaxies. The other suggests the universe is even bigger than that: the universe contains a trillion trillion galaxies. We haven’t yet taken any scientific measurements that could distinguish between these two theories, so each is an equally good explanation for the world as we see it. But wait… we do have some evidence on the question: we exist! If we chose 1/3 in the coin tossing case, by analogy we should say that the theory which results in a trillion trillion galaxies is a trillion times more likely than that which only implies a trillion galaxies. The situation is exactly the same: observers will more often be right if they choose beliefs which imply that many such observers exist, in this case the belief that the universe is really really big.
As it happens we do have just such a theoretical debate about the nature of the universe! One interpretation of results in quantum physics says that the universe is constantly splitting into different worlds:
In many-worlds, the subjective appearance of wavefunction collapse is explained by the mechanism of quantum decoherence. By decoherence, many-worlds claims to resolve all of the correlation paradoxes of quantum theory, such as the EPR paradox and Schrödinger’s cat, since every possible outcome of every event defines or exists in its own “history” or “world”. In layman’s terms, there is a very large—perhaps infinite—number of universes, and everything that could possibly have happened in our past, but didn’t, has occurred in the past of some other universe or universes.
If the many worlds interpretation is correct, the universe is much much much bigger than in any other interpretation of quantum physics. If you are a ‘thirder’ you should now assign a higher probability to the many worlds interpretation than you did before in proportion to how many more observers would exist in the ‘many worlds’ universe than in the ‘single world’ universe.
Detractors of the Self-Indicating Assumption such as Nick Bostrum consider this the ‘reductio ad absurdum’ which shows it must be somehow wrong. As far as I can tell, this is baseless. The only reason they have for rejecting the conclusion is that it seems absurd on its face, but the ‘argument from personal incredulity’ is a weak one, especially on a topic like anthropics which humans did not evolve to think about easily.
If we had good evidence that the universe was actually small, this line of argument might be in conflict. But even before developing this theory we had good reason to think the universe was huge, and other reasons to think it could be even bigger than that! Both theory and experiment neatly point in the same direction: the universe is even bigger than you can imagine.
Join the debate about SIA Katja Grace has kindly started over at Meteuphoric.