The paths followed by probability are not the paths of causal influence, but the paths of logical implication, which run in both directions.
Yep, that was pretty dumb. Thanks for being gentle with me.
However, I still don’t understand what’s wrong with my conclusion in your version of Sleeping Beauty. Upon waking, Sleeping Beauty (whichever copy of her) doesn’t observe anything (colored stones or otherwise) correlated with the result of the coin flip. So it seems she has to stick with her original probability of tails having been flipped, 1⁄2.
Next, out of curiosity, if you had participated in my red/green thought experiment in real life, how would you anticipate if you woke up in a red room (not how would you bet, because I think IRL you’d probably care about copies of you)? I just can’t even physically bring myself to imagine seeing 9,999 copies of me coming out of their respective rooms and telling me they saw red, too, when I had been so confident beforehand that this very situation would not happen. Are you anticipating in the same way as me here?
Finally, let’s pull out the anthropic version of your stones in a bag experiment. Let’s say someone flips an unbiased coin; if it comes up heads, you are knocked out and wake up in a white room, while if it comes up tails, you are knocked out, then copied, and one of you wakes up in a white room and the other wakes up in a black room. Let’s just say the person in each room (or in just the white room if that’s the only one involved) is asked to guess whether the coin came up heads or tails. Let’s also say, for whatever reason, the person has resolved to, if ey wakes up in the white room, guess heads. If ey wakes up in the black room, ey won’t be guessing, ey’ll just be right. Now, if we repeat this experiment multiple times, with different people, it will turn out that, looking at all of the different people (/copies) that actually did wake up in white rooms, it turns out that exactly half of them will have guessed right. Right now I’m just talking about watching this experiment many times from the outside. In fact, it doesn’t matter with what probability the person resolves to guess heads if ey wakes up in the white room—this result holds (that around half of the guesses from white rooms will be correct, in the long run).
Now, given all of that, here’s how I would reason, from the inside of this experiment, if we’re doing log scores in utils (if for some reason I didn’t care about copies of me, which IRL I would) for a probability of heads. Please tell me if you’d reason differently, and why:
In a black room, duh. So let’s say I wake up in a white room. I’d say, well, I only want to maximize my utility. The only way I can be sure to uniquely specify myself, now that I might have been copied, is to say that I am “notsonewuser-in-a-white-room”. Saying “notsonewuser” might not cut it anymore. Historically, when I’ve watched this experiment, “person-in-a-white-room” guesses the coin flip correctly half of the time, no matter what strategy ey has used. So I don’t think I can do better than to say 1⁄2. So I say 1⁄2 and get −1 util (as opposed to an expected −1.08496… utils which I’ve seen historically hold up when I look at all the people in white rooms who have said a 2⁄3 probability of heads).
Now I also need to explain why I think this differs from the obvious situation you brought up (obvious in that the answer was obvious, not in that it wasn’t a good point to make, I think it definitely was!). For one thing, looking historically at people who pick out white stones, they have been in heads-world 2⁄3 of the time. I don’t seem to have any other coherent answer for the difference, though, to be honest (and I’ve already spent hours thinking about this stuff today, and I’m tired). So my reduction’s not quite done, but given the points I’ve made here, I don’t think yours is, either. Maybe you can see flaws in my reasoning, though. Please let me know if you do.
EDIT: I think I figured out the difference. In the situation where you are simply reaching into a bag, the event “I pull out a white stone.” is well defined. In the situation in which you are cloned, the event “I wake up in a white room.” is only well-defined when it is interpreted as “Someone who subjectively experiences being me wakes up in a white room.”, and waking up in a black room is not evidence against the truth of this statement, whereas pulling out a black stone is pretty much absolute evidence that you did not pull out a white stone.
Yep, that was pretty dumb. Thanks for being gentle with me.
However, I still don’t understand what’s wrong with my conclusion in your version of Sleeping Beauty. Upon waking, Sleeping Beauty (whichever copy of her) doesn’t observe anything (colored stones or otherwise) correlated with the result of the coin flip. So it seems she has to stick with her original probability of tails having been flipped, 1⁄2.
Next, out of curiosity, if you had participated in my red/green thought experiment in real life, how would you anticipate if you woke up in a red room (not how would you bet, because I think IRL you’d probably care about copies of you)? I just can’t even physically bring myself to imagine seeing 9,999 copies of me coming out of their respective rooms and telling me they saw red, too, when I had been so confident beforehand that this very situation would not happen. Are you anticipating in the same way as me here?
Finally, let’s pull out the anthropic version of your stones in a bag experiment. Let’s say someone flips an unbiased coin; if it comes up heads, you are knocked out and wake up in a white room, while if it comes up tails, you are knocked out, then copied, and one of you wakes up in a white room and the other wakes up in a black room. Let’s just say the person in each room (or in just the white room if that’s the only one involved) is asked to guess whether the coin came up heads or tails. Let’s also say, for whatever reason, the person has resolved to, if ey wakes up in the white room, guess heads. If ey wakes up in the black room, ey won’t be guessing, ey’ll just be right. Now, if we repeat this experiment multiple times, with different people, it will turn out that, looking at all of the different people (/copies) that actually did wake up in white rooms, it turns out that exactly half of them will have guessed right. Right now I’m just talking about watching this experiment many times from the outside. In fact, it doesn’t matter with what probability the person resolves to guess heads if ey wakes up in the white room—this result holds (that around half of the guesses from white rooms will be correct, in the long run).
Now, given all of that, here’s how I would reason, from the inside of this experiment, if we’re doing log scores in utils (if for some reason I didn’t care about copies of me, which IRL I would) for a probability of heads. Please tell me if you’d reason differently, and why:
In a black room, duh. So let’s say I wake up in a white room. I’d say, well, I only want to maximize my utility. The only way I can be sure to uniquely specify myself, now that I might have been copied, is to say that I am “notsonewuser-in-a-white-room”. Saying “notsonewuser” might not cut it anymore. Historically, when I’ve watched this experiment, “person-in-a-white-room” guesses the coin flip correctly half of the time, no matter what strategy ey has used. So I don’t think I can do better than to say 1⁄2. So I say 1⁄2 and get −1 util (as opposed to an expected −1.08496… utils which I’ve seen historically hold up when I look at all the people in white rooms who have said a 2⁄3 probability of heads).
Now I also need to explain why I think this differs from the obvious situation you brought up (obvious in that the answer was obvious, not in that it wasn’t a good point to make, I think it definitely was!). For one thing, looking historically at people who pick out white stones, they have been in heads-world 2⁄3 of the time. I don’t seem to have any other coherent answer for the difference, though, to be honest (and I’ve already spent hours thinking about this stuff today, and I’m tired). So my reduction’s not quite done, but given the points I’ve made here, I don’t think yours is, either. Maybe you can see flaws in my reasoning, though. Please let me know if you do.
EDIT: I think I figured out the difference. In the situation where you are simply reaching into a bag, the event “I pull out a white stone.” is well defined. In the situation in which you are cloned, the event “I wake up in a white room.” is only well-defined when it is interpreted as “Someone who subjectively experiences being me wakes up in a white room.”, and waking up in a black room is not evidence against the truth of this statement, whereas pulling out a black stone is pretty much absolute evidence that you did not pull out a white stone.