So in this case, I agree that like if this experiment is repeated multiple times and every Sleeping Beauty version created answered tails, the reference class of Sleeping Beauty agents would have many more correct answers than if the experiment is repeated many times and every sleeping Beauty created answered heads.
I think there’s something tangible here and I should reflect on it.
I separately think though that if the actual outcome of each coin flip was recorded, there would be a roughly equal distribution between heads and tails.
And when I was thinking through the question before it was always about trying to answer a question regarding the actual outcome of the coin flip and not what strategy maximises monetary payoffs under even bets.
While I do think that like betting odds isn’t convincing re: actual probabilities because you can just have asymmetric payoffs on equally probable mutually exclusive and jointly exhaustive events, the “reference class of agents being asked this question” seems like a more robust rebuttal.
I want to take some time to think on this.
Strong up voted because this argument actually/genuinely makes me think I might be wrong here.
I separately think though that if the actual outcome of each coin flip was recorded, there would be a roughly equal distribution between heads and tails.
Importantly, this is counting each coinflip as the “experiment”, whereas the above counts each awakening as the “experiment”. It’s okay that different experiments would see different outcome frequencies.
If you record the moments when the outside observer sees the coin landing, you will get 1⁄2.
If you record the moments when the Sleeping Beauty, right after making her bet, is told the actual outcome, you will get 1⁄3.
So we get 1⁄2 by identifying with the outside observer, but he is not the one who was asked in this experiment.
Unless you change the rules so that the Sleeping Beauty is only rewarded for the correct bet at the end of the week, and will only get one reward even if she made two (presumably identical) bets. In that case, recording the moment when the Sleeping Beauty gets the reward or not, you will again get 1⁄2.
I separately think though that if the actual outcome of each coin flip was recorded, there would be a roughly equal distribution between heads and tails.
What I’d say is that this corresponds to the question, “someone tells you they’re running the Sleeping Beauty experiment and just flipped a coin; what’s the probability that it’s heads?”. Difference reference class, different distribution; probability now is 0.5. But this is different from the original question, where we are Sleeping Beauty.
Harth’s framing was presented as an argument re: the canonical Sleeping Beauty problem.
And the question I need to answer is: “should I accept Harth’s frame?”
I am at least convinced that it is genuinely a question about how we define probability.
There is still a disconnect though.
While I agree with the frequentist answer, it’s not clear to me how to backgpropagate this in a Bayesian framework.
Suppose I treat myself as identical to all other agents in the reference class.
I know that my reference class will do better if we answer “tails” when asked about the outcome of the coin toss.
But it’s not obvious to me that there is anything to update from when trying to do a Bayesian probability calculation.
There being many more observers in the tails world to me doesn’t seem to alter these probabilities at all:
P(waking up)
P(being asked questions)
P(...)
By stipulation my observational evidence is the same in both cases.
And I am not compelled by assuming I should be randomly sampled from all observers.
There are many more versions of me in this other world does not by itself seem to raise the probability of me witnessing the observational evidence since by stipulation all versions of me witness the same evidence.
So in this case, I agree that like if this experiment is repeated multiple times and every Sleeping Beauty version created answered tails, the reference class of Sleeping Beauty agents would have many more correct answers than if the experiment is repeated many times and every sleeping Beauty created answered heads.
I think there’s something tangible here and I should reflect on it.
I separately think though that if the actual outcome of each coin flip was recorded, there would be a roughly equal distribution between heads and tails.
And when I was thinking through the question before it was always about trying to answer a question regarding the actual outcome of the coin flip and not what strategy maximises monetary payoffs under even bets.
While I do think that like betting odds isn’t convincing re: actual probabilities because you can just have asymmetric payoffs on equally probable mutually exclusive and jointly exhaustive events, the “reference class of agents being asked this question” seems like a more robust rebuttal.
I want to take some time to think on this.
Strong up voted because this argument actually/genuinely makes me think I might be wrong here.
Much less confident now, and mostly confused.
Importantly, this is counting each coinflip as the “experiment”, whereas the above counts each awakening as the “experiment”. It’s okay that different experiments would see different outcome frequencies.
Yes.
If you record the moments when the outside observer sees the coin landing, you will get 1⁄2.
If you record the moments when the Sleeping Beauty, right after making her bet, is told the actual outcome, you will get 1⁄3.
So we get 1⁄2 by identifying with the outside observer, but he is not the one who was asked in this experiment.
Unless you change the rules so that the Sleeping Beauty is only rewarded for the correct bet at the end of the week, and will only get one reward even if she made two (presumably identical) bets. In that case, recording the moment when the Sleeping Beauty gets the reward or not, you will again get 1⁄2.
What I’d say is that this corresponds to the question, “someone tells you they’re running the Sleeping Beauty experiment and just flipped a coin; what’s the probability that it’s heads?”. Difference reference class, different distribution; probability now is 0.5. But this is different from the original question, where we are Sleeping Beauty.
My current position now is basically: