Ah, so I’ve reinvented the Lewis model. And I suppose that means I’ve inherited its problem where being told that today is Monday makes me think the coin is most likely heads. Oops. And I was just about to claim that there are no contradictions. Sigh.
Okay, I’m starting to understand your claim. To assign a number to P(today is Monday) we basically have two choices. We could just Make Stuff Up and say that it’s 53% or whatever. Or we could at least attempt to do Actual Math. And if our attempt at actual math is coherent enough, then there’s an implicit probability model lurking there, which we can then try to reverse engineer, similar to how you found the Lewis model lurking just beneath the surface of my attempt at math. And once the model is in hand, we can start deriving consequences from it, and Io and behold, before long we have a contradiction, like the Lewis model claiming we can predict the result of a coin flip that hasn’t even happened yet just because we know today is Monday.
And I see now why I personally find the Lewis model so tempting… I was trying to find “small” perturbations of the experiment where “today is Monday” clearly has a well defined probability. But I kept trying use Rare Events to do it, and these change the problem even if the Rare Event is not Observed. (Like, “supposing that my house gets hit by a tornado tomorrow, what is the probability that today is Monday” is fine. Come to think of it, that doesn’t follow Lewis model. Whatever, it’s still fine.)
As for why I find this uncomfortable: I knew that not any string of English words gets a probability, but I was naïve enough to think that all statements that are either true or false get one. And in particular I was hoping they this sequence of posts which kept saying “don’t worry about anthropics, just be careful with the basics and you’ll get the right answer” would show how to answer all possible variations of these “sleep study” questions… instead it turns out that it answers half the questions (the half that ask about the coin) while the other half is shown to be hopeless… and the reason why it’s hopeless really does seem to have an anthropics flavor to it.
I knew that not any string of English words gets a probability, but I was naïve enough to think that all statements that are either true or false get one.
Well, I think this one is actually correct. But, as I said in the previous comment, the statement “Today is Monday” doesn’t actually have a coherent truth value throughout the probability experiment. It’s not either True or False. It’s either True or True and False at the same time!
I was hoping they this sequence of posts which kept saying “don’t worry about anthropics, just be careful with the basics and you’ll get the right answer” would show how to answer all possible variations of these “sleep study” questions… instead it turns out that it answers half the questions (the half that ask about the coin) while the other half is shown to be hopeless… and the reason why it’s hopeless really does seem to have an anthropics flavor to it.
We can answer every coherently formulated question. Everything that is formally defined has an answer Being careful with the basics allows to understand which question is coherent and which is not. This is the same principle as with every probability theory problem.
Consider Sleeping-Beauty experiment without memory loss. There, the event Monday xor Tuesday also can’t be said to always happen. And likewise “Today is Monday” also doesn’t have a stable truth value throughout the whole experiment.
Once again, we can’t express Beauty’s uncertainty between the two days using probability theory. We are just not paying attention to it because by the conditions of the experiment, the Beauty is never in such state of uncertainty. If she remembers a previous awakening then it’s Tuesday, if she doesn’t—then it’s Monday.
All the pieces of the issue are already present. The addition of memory loss just makes it’s obvious that there is the problem with our intuition.
Re: no coherent “stable” truth value: indeed. But still… if she wonders out loud “what day is it?” at the very moment she says that, it has an answer. An experimenter who overhears her knows the answer. It seems to me that you “resolve” this tension is that the two of them are technically asking a different question, even though they are using the same words. But still… how surprised should she be if she were to learn that today is Monday? It seems that taking your stance to its conclusion, the answer would be “zero surprise: she knew for sure she would wake up on Monday so no need to be surprised it happened”
And even if she were to learn that the coin landed tails, so she knows that this is just one of a total of two awakenings, she should have zero surprise upon learning the day of the week, since she now knows both awakenings must happen. Which seems to violate conservation of expected evidence, except you already said that the there’s no coherent probabilities here for that particular question, so that’s fine too.
This makes sense, but I’m not used to it. For instance, I’m used to these questions having the same answer:
P(today is Monday)?
P(today is Monday | the sleep lab gets hit by a tornado)
Yet here, the second question is fine (assuming tornadoes are rare enough that we can ignore the chance of two on consecutive days) while the first makes no sense because we can’t even define “today”
It makes sense but it’s very disorienting, like incompleteness theorem level of disorientation or even more
indeed. But still… if she wonders out loud “what day is it?” at the very moment she says that, it has an answer.
There is no “but”. As long as the Beauty is unable to distinguish between Monday and Tuesday awakenings, as long as the decision process which leads her to say the phrase “what day is it” works the same way, from her perspective there is no one “very moment she says that”. On Tails, there are two different moments when she says this, and the answer is different for them. So there is no answer for her
An experimenter who overhears her knows the answer. It seems to me that you “resolve” this tension is that the two of them are technically asking a different question, even though they are using the same words
Yes, you are correct. From the position of the experimenter, who knows which day it is, or who is hired to work only on one random day this is a coherent question with an actual answer. The words we use are the same but mathematical formalism is different.
For an experimenter who knows that it’s Monday the probability that today is Monday is simply:
P(Monday|Monday) = 1
For an experimenter who is hired to work only on one random day it is:
P(Monday|Monday xor Tuesday) = 1⁄2
But still… how surprised should she be if she were to learn that today is Monday? It seems that taking your stance to its conclusion, the answer would be “zero surprise: she knew for sure she would wake up on Monday so no need to be surprised it happened”
And even if she were to learn that the coin landed tails, so she knows that this is just one of a total of two awakenings, she should have zero surprise upon learning the day of the week, since she now knows both awakenings must happen.
Completely correct. Beauty knew that she would be awaken on Monday either way and so she is not surprised. This is a standard thing with non-mutually exclusive events. Consider this:
A coin is tossed and you are put to sleep. On Heads there will be a red ball in your room. On Tails there will be a red and a blue ball in your room. How surprised should you be to find a red ball in your room?
Which seems to violate conservation of expected evidence, except you already said that the there’s no coherent probabilities here for that particular question, so that’s fine too.
The appearance of violation of conservation of expected evidence comes from the belief that awakening on Monday and on Tuesday are mutually exclusive, while they are, in fact sequential.
This makes sense, but I’m not used to it. For instance, I’m used to these questions having the same answer:
P(today is Monday)?
P(today is Monday | the sleep lab gets hit by a tornado)
Yet here, the second question is fine (assuming tornadoes are rare enough that we can ignore the chance of two on consecutive days) while the first makes no sense because we can’t even define “today”
It makes sense but it’s very disorienting, like incompleteness theorem level of disorientation or even more
I completely understand. It is counterintuitive because evolution didn’t prepare us to deal with situations where an experience is repeated the same while we receive memory loss. As I write in the post:
If I forget what is the current day of the week in my regular life, well, it’s only natural to start from a 1⁄7 prior per day and work from there. I can do it because the causal process that leads to me forgetting such information can be roughly modeled as a low probability occurrence which can happen to me at any day.
It wouldn’t be the case, if I was guaranteed to also forget the current day of the week on the next 6 days as well, after I forgot it on the first one. This would be a different causal process, with different properties—causation between forgetting—and it has to be modeled differently. But we do not actually encounter such situations in everyday life, and so our intuition is caught completely flat footed by them.
The whole paradox arises from this issue with our intuition, and just like with incompleteness theorem (thanks for the flattering comparison, btw), what we need to do now is to re-calibrate our intuitions, make it more accustomed to the truth, preserved by the math, instead of trying to fight it.
Ah, so I’ve reinvented the Lewis model. And I suppose that means I’ve inherited its problem where being told that today is Monday makes me think the coin is most likely heads. Oops. And I was just about to claim that there are no contradictions. Sigh.
Okay, I’m starting to understand your claim. To assign a number to P(today is Monday) we basically have two choices. We could just Make Stuff Up and say that it’s 53% or whatever. Or we could at least attempt to do Actual Math. And if our attempt at actual math is coherent enough, then there’s an implicit probability model lurking there, which we can then try to reverse engineer, similar to how you found the Lewis model lurking just beneath the surface of my attempt at math. And once the model is in hand, we can start deriving consequences from it, and Io and behold, before long we have a contradiction, like the Lewis model claiming we can predict the result of a coin flip that hasn’t even happened yet just because we know today is Monday.
And I see now why I personally find the Lewis model so tempting… I was trying to find “small” perturbations of the experiment where “today is Monday” clearly has a well defined probability. But I kept trying use Rare Events to do it, and these change the problem even if the Rare Event is not Observed. (Like, “supposing that my house gets hit by a tornado tomorrow, what is the probability that today is Monday” is fine. Come to think of it, that doesn’t follow Lewis model. Whatever, it’s still fine.)
As for why I find this uncomfortable: I knew that not any string of English words gets a probability, but I was naïve enough to think that all statements that are either true or false get one. And in particular I was hoping they this sequence of posts which kept saying “don’t worry about anthropics, just be careful with the basics and you’ll get the right answer” would show how to answer all possible variations of these “sleep study” questions… instead it turns out that it answers half the questions (the half that ask about the coin) while the other half is shown to be hopeless… and the reason why it’s hopeless really does seem to have an anthropics flavor to it.
Well, I think this one is actually correct. But, as I said in the previous comment, the statement “Today is Monday” doesn’t actually have a coherent truth value throughout the probability experiment. It’s not either True or False. It’s either True or True and False at the same time!
We can answer every coherently formulated question. Everything that is formally defined has an answer Being careful with the basics allows to understand which question is coherent and which is not. This is the same principle as with every probability theory problem.
Consider Sleeping-Beauty experiment without memory loss. There, the event Monday xor Tuesday also can’t be said to always happen. And likewise “Today is Monday” also doesn’t have a stable truth value throughout the whole experiment.
Once again, we can’t express Beauty’s uncertainty between the two days using probability theory. We are just not paying attention to it because by the conditions of the experiment, the Beauty is never in such state of uncertainty. If she remembers a previous awakening then it’s Tuesday, if she doesn’t—then it’s Monday.
All the pieces of the issue are already present. The addition of memory loss just makes it’s obvious that there is the problem with our intuition.
Re: no coherent “stable” truth value: indeed. But still… if she wonders out loud “what day is it?” at the very moment she says that, it has an answer. An experimenter who overhears her knows the answer. It seems to me that you “resolve” this tension is that the two of them are technically asking a different question, even though they are using the same words. But still… how surprised should she be if she were to learn that today is Monday? It seems that taking your stance to its conclusion, the answer would be “zero surprise: she knew for sure she would wake up on Monday so no need to be surprised it happened”
And even if she were to learn that the coin landed tails, so she knows that this is just one of a total of two awakenings, she should have zero surprise upon learning the day of the week, since she now knows both awakenings must happen. Which seems to violate conservation of expected evidence, except you already said that the there’s no coherent probabilities here for that particular question, so that’s fine too.
This makes sense, but I’m not used to it. For instance, I’m used to these questions having the same answer:
P(today is Monday)?
P(today is Monday | the sleep lab gets hit by a tornado)
Yet here, the second question is fine (assuming tornadoes are rare enough that we can ignore the chance of two on consecutive days) while the first makes no sense because we can’t even define “today”
It makes sense but it’s very disorienting, like incompleteness theorem level of disorientation or even more
There is no “but”. As long as the Beauty is unable to distinguish between Monday and Tuesday awakenings, as long as the decision process which leads her to say the phrase “what day is it” works the same way, from her perspective there is no one “very moment she says that”. On Tails, there are two different moments when she says this, and the answer is different for them. So there is no answer for her
Yes, you are correct. From the position of the experimenter, who knows which day it is, or who is hired to work only on one random day this is a coherent question with an actual answer. The words we use are the same but mathematical formalism is different.
For an experimenter who knows that it’s Monday the probability that today is Monday is simply:
P(Monday|Monday) = 1
For an experimenter who is hired to work only on one random day it is:
P(Monday|Monday xor Tuesday) = 1⁄2
Completely correct. Beauty knew that she would be awaken on Monday either way and so she is not surprised. This is a standard thing with non-mutually exclusive events. Consider this:
A coin is tossed and you are put to sleep. On Heads there will be a red ball in your room. On Tails there will be a red and a blue ball in your room. How surprised should you be to find a red ball in your room?
The appearance of violation of conservation of expected evidence comes from the belief that awakening on Monday and on Tuesday are mutually exclusive, while they are, in fact sequential.
I completely understand. It is counterintuitive because evolution didn’t prepare us to deal with situations where an experience is repeated the same while we receive memory loss. As I write in the post:
The whole paradox arises from this issue with our intuition, and just like with incompleteness theorem (thanks for the flattering comparison, btw), what we need to do now is to re-calibrate our intuitions, make it more accustomed to the truth, preserved by the math, instead of trying to fight it.
Thanks :) the recalibration may take a while… my intuition is still fighting ;)