What I’m arguing against is apparently neither SIA or SSA. I made a mistake. Are we arguing about what I originally intended to, or about my statement that they both predict 1/3?
My intent was to argue the, if you only update on the existence of an observer, rather than anything about how unlikely it is to be them, the probability will work out the same.
If you would like to discuss why SIA and SSA give the same result:
For simplicity, we’ll assume 1 trillion observation days outside the experiment.
SIA: Sleeping beauty wakes up in this experiment. There are 2 trillion 3 possible observers, three of which wake up here. Of them, one woke up in a universe with heads, and the other in a universe with tails. The probability of being the one with heads is 1⁄3.
SSA: Sleeping beauty has an even prior, so the odds ratio of heads to tails is 1:1. She then wakes up in this experiment. If the coin landed on heads, there’s a 1⁄1 trillion 1 chance of this. If the coin landed on tails, there’s a 2⁄1 trillion 2 chance of this. This is an odds ratio of 500,000,000,001:1,000,000,000,001 for heads. Multiplying this by 1:1 yields 500,000,000,001:1,000,000,000,001. The total probability of heads is 1⁄3 + 2*10^-13.
Your problem seems to be updating on the fact that she’s in the experiment without taking into account that this is about twice as likely if the coin landed on heads.
...huh. You have a point. I’ll have to think about this for a bit, but it seems right, and if this is what you’ve been trying to get at this whole time I think everyone may have misunderstood you.
What I’m arguing against is apparently neither SIA or SSA. I made a mistake. Are we arguing about what I originally intended to, or about my statement that they both predict 1/3?
My intent was to argue the, if you only update on the existence of an observer, rather than anything about how unlikely it is to be them, the probability will work out the same.
If you would like to discuss why SIA and SSA give the same result:
For simplicity, we’ll assume 1 trillion observation days outside the experiment.
SIA: Sleeping beauty wakes up in this experiment. There are 2 trillion 3 possible observers, three of which wake up here. Of them, one woke up in a universe with heads, and the other in a universe with tails. The probability of being the one with heads is 1⁄3.
SSA: Sleeping beauty has an even prior, so the odds ratio of heads to tails is 1:1. She then wakes up in this experiment. If the coin landed on heads, there’s a 1⁄1 trillion 1 chance of this. If the coin landed on tails, there’s a 2⁄1 trillion 2 chance of this. This is an odds ratio of 500,000,000,001:1,000,000,000,001 for heads. Multiplying this by 1:1 yields 500,000,000,001:1,000,000,000,001. The total probability of heads is 1⁄3 + 2*10^-13.
Your problem seems to be updating on the fact that she’s in the experiment without taking into account that this is about twice as likely if the coin landed on heads.
...huh. You have a point. I’ll have to think about this for a bit, but it seems right, and if this is what you’ve been trying to get at this whole time I think everyone may have misunderstood you.