I’m not sure what you mean by “Beauty’s expected experience … on Wednesday”. There are two possibilities. On Sunday, she will expect that they have equal probabilities. But that says nothing more than that on Sunday she expects Heads and Tails to be equally likely. When woken on Monday or Tuesday, here expectation of what Wednesday will be like isn’t necessarily the same.
I also can’t figure out what you mean by saying that 1⁄3 is “absolutely correct”, while also saying that there could be another answer.
What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday? If it’s 1⁄2 what is the reason for the change from 1/3? The fact that she has forgotten the information about her awakening during the experiment?
Suppose SB wakes up on Wednesday. She is told that it’s Wednesday and then left alone to dress before she will be asked for the last time to guess Heads or Tails. She doesn’t remember any of her awakenings and her credence for Heads is once again 1⁄2. However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1⁄3. Should she expect to successfully guess Tails with 2⁄3 probability?
Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn’t know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?
What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday? If it’s 1⁄2 what is the reason for the change from 1/3? The fact that she has forgotten the information about her awakening during the experiment?
Because of the memory erasure, I think it’s best to regard Beauty on Monday, on Tuesday (if woken), and on Wednesday as different people, all “descended” from Beauty on Sunday. But if you want to think of them as the same person, the change from 1⁄3 to 1⁄2 is a consequence of the new information from being told that it is Wednesday. The same shift happens if she’s told that it is Monday. (If she’s told it is Tuesday, she of course shifts to probability 1 for Tails.)
In the standard setup, by the way, I think it’s implicit that on Monday and Tuesday awakenings, Beauty is told that it is not Wednesday (ie, that it is either Monday or Tuesday).
Suppose SB wakes up on Wednesday. She is told that it’s Wednesday and then left alone to dress before she will be asked for the last time to guess Heads or Tails. She doesn’t remember any of her awakenings and her credence for Heads is once again 1⁄2. However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1⁄3. Should she expect to successfully guess Tails with 2⁄3 probability?
Assuming that successfully hiding a piece of paper is an unlikely event, she should think the probability of Heads is 1⁄3 simply because that makes a hidden piece of paper twice as likely (since then her past self will have had two chances to do this). The message on the paper is irrelevant (assuming it doesn’t contain any new arguments that she hadn’t thought of on Sunday).
Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn’t know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?
That’s an interesting scenario. My argument for why Beauty when not told it is Wednesday should think Tails is twice as likely as Heads is that with Tails (hence two awakenings) the probability of anyone experiencing what she is (eg, exact position of fly seen on the wall, stray thought about how good omelettes taste, …) is twice what it is with Heads (one awakening), so standard Bayesian updating makes Tails twice as likely. When woken, told it is Wednesday, and having memories of her last awakening, however, the probability of her experience (including memories) is the same for Heads or Tails. The two awakenings with Tails don’t double the probability of her remembered experience being whatever it is (along with knowing it is Wednesday), since whatever she experienced on Monday is forgotten. So her probability for Heads should be 1⁄2.
the change from 1⁄3 to 1⁄2 is a consequence of the new information from being told that it is Wednesday.
How does it work? Especially in the scenario where SB doesn’t loose her memory of the last awakening. What kind of Bayesian update is it? Why are coins less likely to turn out to be Tails On Wednesday than on Monday or Tuesday? Does the universe retcon the result of the coin toss somehow?
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.
Ape in the coat: “What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday?”
The snarky answer is “irrelevant.” The assessment of her credence, that she gives on Monday or Tuesday, is based on the information that it is either Monday or Tuesday within the waking schedule described in the experiment. She is not responsible for incorporating false information into her credence.
One thing that seems to get lost in many threads about this problem, is that probability and/or credence is a function of your knowledge about the state of the system. It is not a property of the system itself. You are trying to make it a property of the system.
Halfers are most guilty of not understanding this, since their arguments are based on there being no change to the system despite an obvious change to the knowledge (I’m ignoring how to treat that knowledge). Thirders recognize the change, but keep trying to cast the difference in terms of episiotomy when it really is much simpler.
“However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1⁄3. Should she expect to successfully guess Tails with 2⁄3 probability?”
No, since her knowledge is different now than it was then.
“Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn’t know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?”
If I understand this correctly, she knows that it is Wednesday. Her credence is 1⁄2. She can, at the same time on Sunday or Wednesday, make an assessment of the state of her knowledge on Monday or Tuesday, and recognize how it is different. In other words, “my credence now is 1⁄2, but my credence then should be 1⁄3.” The paradox you seem to be fishing for does not exist.
One thing that seems to get lost in many threads about this problem, is that probability and/or credence is a function of your knowledge about the state of the system. It is not a property of the system itself. You are trying to make it a property of the system.
Your knowledge about the state of the system can either correctly represent the state of the system or not. If you believe that the probability that the coin is Tails is 2⁄3, while on a repeated experiment about 50% times the coin is Heads that means that your probability estimate is not correct. Of course you can declare that your credence is answering some other question and this is fine. But there is a potential for misunderstanding—people can mistakenly think that they can actually guess the result of the coin toss better than chance as if they received some relevant evidence.
Whether your knowledge correctly represents the state of the system, or not, is irrelevant. Your credence is based on your knowledge. If they lie to SB and wake her twice after Heads, and three times after Tails? But still tell here it is once or twice? A thirder’s credence should still be 1⁄3.
And those who guess would not be considered the rational probability agent that some versions insist SB must be.
The correct answer is 1⁄3. See my answer to this question.
Whether your knowledge correctly represents the state of the system, or not, is irrelevant
Well, when you explicitly acknowledge that “2/3 credence for Tails” doesn’t correspond to Tails being the result of the coin toss in 2⁄3 of experiments where the beauty is thinking this way, we have cleared the possible misunderstanding. You are talking about the motte not the bailey. We can still argue about whose definitions are better, and I’d like to make a few points about it. But essentially the crux is resolved.
If they lie to SB and wake her twice after Heads, and three times after Tails?
In such mind experiments the possibility of lies is usually stated explicitly. Like in the riddle of three idols. And people do not usually claim that if the person trusted the lying idol their belief is “correct” and it’s not their fault they got lied to.
When the possibility of a lie wasn’t explicitly stated but the lie happened, then your point is valid. We can say that the credence corresponded to a different experiment setting, thus in this sense can still be correct, even if it’s not correct in the new experimental setting.
However, this is absolutely not the case here. No one is lying to the beauty. The experiment is going exactly is stated. So if beauty’s knowledge doesn’t represent the state of the system, it means she made some incorrect inference from her evidence which is her failure as a rational agent.
The correct answer is 1⁄3.
When you declare that map representing the territory is irrelevant it’s not clear how “correct” is a meaningful term anymore. I suppose you mean that it corresponds to your mathematical model. This is true but you can always find some mathematical model that your answer corresponds to. The interesting question is whether this model corresponds to reality.
My point is that SB must have reason to think that she exists in the “Monday or Tuesday” waking schedule, for her to assign a credence to Heads based on that schedule. If she is awake, but has any reason to think she is not in that situation, her credence must take that into account.
You told her that she would be asked for her credence on Monday and maybe on Tuesday. “What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday?” is irrelevant because you are allowing for the case where that is not true yet you want her to believe it is. That is lying to her, in the context of the information you want her to use.
But she can from an opinion. “My credence should be halfer/thirder answer if Wednesday has not yet dawned, or 1⁄2 if it has. Since the cue I was told would happen—being asked for my credence—has not yet occurred, I am uncertain which and so can’t give a more definitive answer.” And if you give that cue on Wednesday, you are lying even if you promised you wouldn’t.
And yes, my mathematical model corresponds to SB’s reality when she is asked for her credence. That is the entire point. If you think otherwise, I’d love to hear an explanation instead of a dissertation that does not apply.
You told her that she would be asked for her credence on Monday and maybe on Tuesday. “What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday?” is irrelevant because you are allowing for the case where that is not true yet you want her to believe it is. That is lying to her, in the context of the information you want her to use.
No lies are necessary. I didn’t have to tell her beforehand that she will be asked any questions at all. Or I could have told her that she will be asked on Monday and Tuesday (if she is awake) without knowing which day it is and then she will be asked on Wednesday, knowing that it is indeed Wednesday. None of it changes the experiment. And even if I didn’t ask her about her credence that the coin is Heads on Wednesday, she still has to have some probability estimate doesn’t she?
The point of the Wednesday question is to highlight, that, what you mean by “credence”, isn’t actually a probability estimate that the coin is Heads. What you are talking about, is the probability that the coin is Heads, weighted by the number of times this question is asked. Which is a meaningful category. But confusing it with the probability that the coin is Heads can be extremely misleading
AINC: “What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday?”
If she has no reason to think this is not one of her awakenings on Monday/Tuesday, then her credence is the same as it would have been then: 1⁄2 if she is a halfer, and 1⁄3 if she is a thirder.
AINC: “If it’s 1⁄2 what is the reason for the change from 1/3?”
The only way it could change from 1⁄3 to 1⁄2, is if she is a thirder and you tell her that it is Wednesday. And the reason it changes is that you changes the state of her information, not because anything about the coin itself has changed.
But if you think her credence should be based on the actual day, even when you didn’t tell her that is was Wednesday, then you have told an implicit lie. You are asking her to formulate a credence based on Monday/Tuesday, but expecting her answer to be consistent with Wednesday.
AINC: The point of the Wednesday question is to highlight, that, what you mean by “credence”, isn’t actually a probability estimate that the coin is Heads.
And the point of my answer, is that it is actually a conditional probability based on an unusual state of information.
I’m not sure what you mean by “Beauty’s expected experience … on Wednesday”. There are two possibilities. On Sunday, she will expect that they have equal probabilities. But that says nothing more than that on Sunday she expects Heads and Tails to be equally likely. When woken on Monday or Tuesday, here expectation of what Wednesday will be like isn’t necessarily the same.
I also can’t figure out what you mean by saying that 1⁄3 is “absolutely correct”, while also saying that there could be another answer.
What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday? If it’s 1⁄2 what is the reason for the change from 1/3? The fact that she has forgotten the information about her awakening during the experiment?
Suppose SB wakes up on Wednesday. She is told that it’s Wednesday and then left alone to dress before she will be asked for the last time to guess Heads or Tails. She doesn’t remember any of her awakenings and her credence for Heads is once again 1⁄2. However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1⁄3. Should she expect to successfully guess Tails with 2⁄3 probability?
Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn’t know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?
What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday? If it’s 1⁄2 what is the reason for the change from 1/3? The fact that she has forgotten the information about her awakening during the experiment?
Because of the memory erasure, I think it’s best to regard Beauty on Monday, on Tuesday (if woken), and on Wednesday as different people, all “descended” from Beauty on Sunday. But if you want to think of them as the same person, the change from 1⁄3 to 1⁄2 is a consequence of the new information from being told that it is Wednesday. The same shift happens if she’s told that it is Monday. (If she’s told it is Tuesday, she of course shifts to probability 1 for Tails.)
In the standard setup, by the way, I think it’s implicit that on Monday and Tuesday awakenings, Beauty is told that it is not Wednesday (ie, that it is either Monday or Tuesday).
Suppose SB wakes up on Wednesday. She is told that it’s Wednesday and then left alone to dress before she will be asked for the last time to guess Heads or Tails. She doesn’t remember any of her awakenings and her credence for Heads is once again 1⁄2. However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1⁄3. Should she expect to successfully guess Tails with 2⁄3 probability?
Assuming that successfully hiding a piece of paper is an unlikely event, she should think the probability of Heads is 1⁄3 simply because that makes a hidden piece of paper twice as likely (since then her past self will have had two chances to do this). The message on the paper is irrelevant (assuming it doesn’t contain any new arguments that she hadn’t thought of on Sunday).
Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn’t know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?
That’s an interesting scenario. My argument for why Beauty when not told it is Wednesday should think Tails is twice as likely as Heads is that with Tails (hence two awakenings) the probability of anyone experiencing what she is (eg, exact position of fly seen on the wall, stray thought about how good omelettes taste, …) is twice what it is with Heads (one awakening), so standard Bayesian updating makes Tails twice as likely. When woken, told it is Wednesday, and having memories of her last awakening, however, the probability of her experience (including memories) is the same for Heads or Tails. The two awakenings with Tails don’t double the probability of her remembered experience being whatever it is (along with knowing it is Wednesday), since whatever she experienced on Monday is forgotten. So her probability for Heads should be 1⁄2.
How does it work? Especially in the scenario where SB doesn’t loose her memory of the last awakening. What kind of Bayesian update is it? Why are coins less likely to turn out to be Tails On Wednesday than on Monday or Tuesday? Does the universe retcon the result of the coin toss somehow?
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.
Ape in the coat: “What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday?”
The snarky answer is “irrelevant.” The assessment of her credence, that she gives on Monday or Tuesday, is based on the information that it is either Monday or Tuesday within the waking schedule described in the experiment. She is not responsible for incorporating false information into her credence.
One thing that seems to get lost in many threads about this problem, is that probability and/or credence is a function of your knowledge about the state of the system. It is not a property of the system itself. You are trying to make it a property of the system.
Halfers are most guilty of not understanding this, since their arguments are based on there being no change to the system despite an obvious change to the knowledge (I’m ignoring how to treat that knowledge). Thirders recognize the change, but keep trying to cast the difference in terms of episiotomy when it really is much simpler.
“However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1⁄3. Should she expect to successfully guess Tails with 2⁄3 probability?”
No, since her knowledge is different now than it was then.
“Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn’t know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?”
If I understand this correctly, she knows that it is Wednesday. Her credence is 1⁄2. She can, at the same time on Sunday or Wednesday, make an assessment of the state of her knowledge on Monday or Tuesday, and recognize how it is different. In other words, “my credence now is 1⁄2, but my credence then should be 1⁄3.” The paradox you seem to be fishing for does not exist.
Your knowledge about the state of the system can either correctly represent the state of the system or not. If you believe that the probability that the coin is Tails is 2⁄3, while on a repeated experiment about 50% times the coin is Heads that means that your probability estimate is not correct. Of course you can declare that your credence is answering some other question and this is fine. But there is a potential for misunderstanding—people can mistakenly think that they can actually guess the result of the coin toss better than chance as if they received some relevant evidence.
I explore this dynamics in my recent post.
Whether your knowledge correctly represents the state of the system, or not, is irrelevant. Your credence is based on your knowledge. If they lie to SB and wake her twice after Heads, and three times after Tails? But still tell here it is once or twice? A thirder’s credence should still be 1⁄3.
And those who guess would not be considered the rational probability agent that some versions insist SB must be.
The correct answer is 1⁄3. See my answer to this question.
Well, when you explicitly acknowledge that “2/3 credence for Tails” doesn’t correspond to Tails being the result of the coin toss in 2⁄3 of experiments where the beauty is thinking this way, we have cleared the possible misunderstanding. You are talking about the motte not the bailey. We can still argue about whose definitions are better, and I’d like to make a few points about it. But essentially the crux is resolved.
In such mind experiments the possibility of lies is usually stated explicitly. Like in the riddle of three idols. And people do not usually claim that if the person trusted the lying idol their belief is “correct” and it’s not their fault they got lied to.
When the possibility of a lie wasn’t explicitly stated but the lie happened, then your point is valid. We can say that the credence corresponded to a different experiment setting, thus in this sense can still be correct, even if it’s not correct in the new experimental setting.
However, this is absolutely not the case here. No one is lying to the beauty. The experiment is going exactly is stated. So if beauty’s knowledge doesn’t represent the state of the system, it means she made some incorrect inference from her evidence which is her failure as a rational agent.
When you declare that map representing the territory is irrelevant it’s not clear how “correct” is a meaningful term anymore. I suppose you mean that it corresponds to your mathematical model. This is true but you can always find some mathematical model that your answer corresponds to. The interesting question is whether this model corresponds to reality.
My point is that SB must have reason to think that she exists in the “Monday or Tuesday” waking schedule, for her to assign a credence to Heads based on that schedule. If she is awake, but has any reason to think she is not in that situation, her credence must take that into account.
You told her that she would be asked for her credence on Monday and maybe on Tuesday. “What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday?” is irrelevant because you are allowing for the case where that is not true yet you want her to believe it is. That is lying to her, in the context of the information you want her to use.
But she can from an opinion. “My credence should be halfer/thirder answer if Wednesday has not yet dawned, or 1⁄2 if it has. Since the cue I was told would happen—being asked for my credence—has not yet occurred, I am uncertain which and so can’t give a more definitive answer.” And if you give that cue on Wednesday, you are lying even if you promised you wouldn’t.
And yes, my mathematical model corresponds to SB’s reality when she is asked for her credence. That is the entire point. If you think otherwise, I’d love to hear an explanation instead of a dissertation that does not apply.
No lies are necessary. I didn’t have to tell her beforehand that she will be asked any questions at all. Or I could have told her that she will be asked on Monday and Tuesday (if she is awake) without knowing which day it is and then she will be asked on Wednesday, knowing that it is indeed Wednesday. None of it changes the experiment. And even if I didn’t ask her about her credence that the coin is Heads on Wednesday, she still has to have some probability estimate doesn’t she?
The point of the Wednesday question is to highlight, that, what you mean by “credence”, isn’t actually a probability estimate that the coin is Heads. What you are talking about, is the probability that the coin is Heads, weighted by the number of times this question is asked. Which is a meaningful category. But confusing it with the probability that the coin is Heads can be extremely misleading
AINC: “What’s Beauty credence for Heads when she wakes on Wednesday and doesn’t remember any of her awakenings on Monday/Tuesday?”
If she has no reason to think this is not one of her awakenings on Monday/Tuesday, then her credence is the same as it would have been then: 1⁄2 if she is a halfer, and 1⁄3 if she is a thirder.
AINC: “If it’s 1⁄2 what is the reason for the change from 1/3?”
The only way it could change from 1⁄3 to 1⁄2, is if she is a thirder and you tell her that it is Wednesday. And the reason it changes is that you changes the state of her information, not because anything about the coin itself has changed.
But if you think her credence should be based on the actual day, even when you didn’t tell her that is was Wednesday, then you have told an implicit lie. You are asking her to formulate a credence based on Monday/Tuesday, but expecting her answer to be consistent with Wednesday.
AINC: The point of the Wednesday question is to highlight, that, what you mean by “credence”, isn’t actually a probability estimate that the coin is Heads.
And the point of my answer, is that it is actually a conditional probability based on an unusual state of information.