Let’s look at the scenario where Beauty remembers her last awakening.
When Beauty is woken and not told it’s Wednesday, she should think Heads has probability 1⁄3, Tails 2⁄3. She knows that if the coin landed Heads, it is Monday, and that she will next wake up on Wednesday without forgetting anything. She knows that if the coin landed Tails, then with probability 1⁄2 it is Monday and her memory will soon be erased, and with probability 1⁄2 it is Tuesday and she will next wake up on Wednesday without forgetting anything.
So waking up on Wednesday without forgetting anything is twice as likely for Heads as for Tails. Hence, if that’s what happens, she should update her probability of Heads from 1⁄3 to to 1⁄2 (ie, odds for Heads change from 1⁄2 to 1, changing by a factor of two due to the likelihood ratio of two).
But what about if instead her memory is erased before a Tuesday awakening? In that case, this instance of Beauty effectively dies, and there’s nothing to say about what her probability of Heads is.
Compare with the situation where someone plays Russian roulette with a revolver that they know has probability 1⁄2 of being empty, and probability 1⁄2 of having bullets in 3 of the 6 positions. If they survive, they should update their probability of the revolver being empty to 2⁄3 (ie, odds for empty change from 1 to 2), since they are twice as likely to survive if it is empty as if it has 3 of 6 positions with bullets.
Similarly, if Beauty mistakenly thinks when woken and told it’s not Wednesday that Heads and Tails both have probability 1⁄2, then if she is woken on Wednesday without forgetting anything, she should by the rules of probability change her probability of Heads to 2⁄3. And that is obviously wrong.
She knows that if the coin landed Heads, it is Monday, and that she will next wake up on Wednesday without forgetting anything. She knows that if the coin landed Tails, then with probability 1⁄2 it is Monday and her memory will soon be erased, and with probability 1⁄2 it is Tuesday and she will next wake up on Wednesday without forgetting anything.
Indeed. This I understand.
So waking up on Wednesday without forgetting anything is twice as likely for Heads as for Tails.
And here I stopped following you. There is exactly one person in her epistemic situation (Waking up on Wednesday remembering the previous awakening) in both Heads and Tails worlds. According to both SSA and SIA no update has to happen.
There are two ways one might try to figure out these probabilities. One is that, in whatever final situation is being considered, you figure out the probability from scratch, as if the question had never occurred to you before. The other is that as you experience things you update your probability for something, according to the the likelihood ratio obtained from what you just observed, and in that way obtain a probability in the final situation.
When figuring out the probability of Heads from scratch on Wednesday, I think everyone agrees that it should be 1⁄2. The only issue from my perspective is that one might think that my argument for Heads having probability 1⁄3 when Beauty is woken before Wednesday still applies, but as I explained above, it doesn’t.
So the question now is whether this is consistent with updating the probability of Heads from a value of 1⁄3 when Beauty is woken before Wednesday, assuming that this instance of beauty does not have her memory erased before being woken on Wednesday. Not having her memory erased and then being woken on Wednesday is twice as likely for Heads as for Tails, so yes, the probability does get updated to 1⁄2, consistent with the “from scratch” method.
Note that it makes no sense to ask how an instance of Beauty whose memory has been erased will “update” her probability of Heads—she doesn’t even remember what “her” (really someone else’s) previous probability of Heads was.
As far as I can tell, you don’t seem to be updating based on the ratio of likelihoods from what is observed, but I don’t know what method you are using. SSA and SIA aren’t usually phrased as methods for updating probabilities over time as new information arrives. And even if they were, it would certainly be controversial to say that they should take precedence over standard Bayesian reasoning.
But in any case, I don’t agree that “there is exactly one person… waking up on Wednesday remembering their previous awakening”. There is of course only one person who is actually woken on Wednesday, but I take it you mean potential persons, who might have been the one woken on Wednesday. There is Beauty on Sunday, who if the coin lands Heads will not have her memory erased at any point and will then be woken on Wednesday. Alternatively, when the coin lands Tails, there is Beauty as woken on Tuesday, who from that point on does not have her memory erased and is woken on Wednesday. The Beauty who is woken on Monday and then has her memory erased is effectively dead.
Let’s look at the scenario where Beauty remembers her last awakening.
When Beauty is woken and not told it’s Wednesday, she should think Heads has probability 1⁄3, Tails 2⁄3. She knows that if the coin landed Heads, it is Monday, and that she will next wake up on Wednesday without forgetting anything. She knows that if the coin landed Tails, then with probability 1⁄2 it is Monday and her memory will soon be erased, and with probability 1⁄2 it is Tuesday and she will next wake up on Wednesday without forgetting anything.
So waking up on Wednesday without forgetting anything is twice as likely for Heads as for Tails. Hence, if that’s what happens, she should update her probability of Heads from 1⁄3 to to 1⁄2 (ie, odds for Heads change from 1⁄2 to 1, changing by a factor of two due to the likelihood ratio of two).
But what about if instead her memory is erased before a Tuesday awakening? In that case, this instance of Beauty effectively dies, and there’s nothing to say about what her probability of Heads is.
Compare with the situation where someone plays Russian roulette with a revolver that they know has probability 1⁄2 of being empty, and probability 1⁄2 of having bullets in 3 of the 6 positions. If they survive, they should update their probability of the revolver being empty to 2⁄3 (ie, odds for empty change from 1 to 2), since they are twice as likely to survive if it is empty as if it has 3 of 6 positions with bullets.
Similarly, if Beauty mistakenly thinks when woken and told it’s not Wednesday that Heads and Tails both have probability 1⁄2, then if she is woken on Wednesday without forgetting anything, she should by the rules of probability change her probability of Heads to 2⁄3. And that is obviously wrong.
Indeed. This I understand.
And here I stopped following you. There is exactly one person in her epistemic situation (Waking up on Wednesday remembering the previous awakening) in both Heads and Tails worlds. According to both SSA and SIA no update has to happen.
There are two ways one might try to figure out these probabilities. One is that, in whatever final situation is being considered, you figure out the probability from scratch, as if the question had never occurred to you before. The other is that as you experience things you update your probability for something, according to the the likelihood ratio obtained from what you just observed, and in that way obtain a probability in the final situation.
When figuring out the probability of Heads from scratch on Wednesday, I think everyone agrees that it should be 1⁄2. The only issue from my perspective is that one might think that my argument for Heads having probability 1⁄3 when Beauty is woken before Wednesday still applies, but as I explained above, it doesn’t.
So the question now is whether this is consistent with updating the probability of Heads from a value of 1⁄3 when Beauty is woken before Wednesday, assuming that this instance of beauty does not have her memory erased before being woken on Wednesday. Not having her memory erased and then being woken on Wednesday is twice as likely for Heads as for Tails, so yes, the probability does get updated to 1⁄2, consistent with the “from scratch” method.
Note that it makes no sense to ask how an instance of Beauty whose memory has been erased will “update” her probability of Heads—she doesn’t even remember what “her” (really someone else’s) previous probability of Heads was.
As far as I can tell, you don’t seem to be updating based on the ratio of likelihoods from what is observed, but I don’t know what method you are using. SSA and SIA aren’t usually phrased as methods for updating probabilities over time as new information arrives. And even if they were, it would certainly be controversial to say that they should take precedence over standard Bayesian reasoning.
But in any case, I don’t agree that “there is exactly one person… waking up on Wednesday remembering their previous awakening”. There is of course only one person who is actually woken on Wednesday, but I take it you mean potential persons, who might have been the one woken on Wednesday. There is Beauty on Sunday, who if the coin lands Heads will not have her memory erased at any point and will then be woken on Wednesday. Alternatively, when the coin lands Tails, there is Beauty as woken on Tuesday, who from that point on does not have her memory erased and is woken on Wednesday. The Beauty who is woken on Monday and then has her memory erased is effectively dead.