The second one looks “obvious” from symmetry considerations but actually formalizing seems harder than expected.
Exactly! I’m glad that you actually engaged with the problem.
The first step is to realize that here “today” can’t mean “Monday xor Tuesday” because such event never happens. On every iteration of experiment both Monday and Tuesday are realized. So we can’t say that the participant knows that they are awakened on Monday xor Tuesday.
Can we say that participant knows that they are awakened on Monday or Tuesday? Sure. As a matter of fact:
P(Monday or Tuesday) = 1
P(Heads|Monday or Tuesday) = P(Heads) = 1⁄2
This works, here probability that the coin is Heads in this iteration of the experiment happens to be the same as what our intuition is telling us P(Heads|Today) is supposed to be, however we still can’t define “Today is Monday”:
P(Monday|Monday or Tuesday) = P(Monday) = 1
Which doesn’t fit our intuition.
How can this be? How can we have a seeminglly well-defined probability for “Today the coin is Heads” but not for “Today is Monday”? Either “Today” is well-defined or it’s not, right? Take some time to think about it.
What do we actually mean when we say that on an awakening the participant supposed to believe that the coin is Heads with 50% probability? Is it really about this day in particular? Or is it about something else?
The answer is: we actually mean, that on any day of the experiment be it Monday or Tuesday the participant is supposed to believe that the coin is Heads with 50% probability. We can not formally specify “Today” in this problem but there is a clever, almost cheating way to specify “Anyday” without breaking anything.
This is not easy. It requires a way to define P(A|B), when P(B) itself is undefined which is unconventional. But, moreover, it requires symmetry. P(Heads|Monday) has to be equal to P(Heads|Tuesday) only then we have a coherent P(Heads|Anyday).
This makes me uncomfortable. From the perspective of sleeping beauty, who just woke up, the statement “today is Monday” is either true or false (she just doesn’t know which one). Yet you claim she can’t meaningfully assign it a probability. This feels wrong, and yet, if I try to claim that the probability is, say, 2⁄3, then you will ask me “in what sample space?” and I don’t know the answer.
What seems clear is that the sample space is not the usual sleeping beauty sample space; it has to run metaphorically “skew” to it somehow.
If the question were “did the coin land on heads” then it’s clear that this is question is of the form “what world am I in?”. Namely, “am I in a world where the coin landed on heads, or not?”
Likewise if we ask “does a Tuesday awakening happen?”… that maps easily to question about the coin, so it’s safe.
But there should be a way to ask about today as well, I think. Let’s try something naive first and see where it breaks.
P(today is Monday | heads) = 100% is fine.
(Or is that tails? I keep forgetting.)
P(today is Monday | tails) = 50% is fine too.
(Or maybe it’s not? Maybe this is where I’m going working? Needs a bit of work but I suspect I could formalize that one if I had to.)
But if those are both fine, we should be able to combine them, like so:
heads and tails are mutually exclusive and one of them must happen, so:
P(today is Monday) =
P(heads) • P(today is Monday | heads) +
P(tails) • P(today is Monday | tails) =
0.5 + .25 = 0.75
Okay, I was expecting to get 2⁄3 here. Odd. More to the point, this felt like cheating and I can’t put my finger on why.
maybe need to think more later
This makes me uncomfortable. From the perspective of sleeping beauty, who just woke up, the statement “today is Monday” is either true or false (she just doesn’t know which one). Yet you claim she can’t meaningfully assign it a probability. This feels wrong, and yet, if I try to claim that the probability is, say, 2⁄3, then you will ask me “in what sample space?” and I don’t know the answer.
Where does the feeling of wrongness come from? Were you under impression that probability theory promised us to always assign some measure to any statement in natural language? It just so happens that most of the time we can construct an appropriate probability space. But the actual rule is about whether or not we can construct a probability space, not whether or not something is a statement in natural language.
Is it really so surprising that a person who is experiencing amnesia and the repetetion of the same experience, while being fully aware of the procedure can’t meaningfully assign credence to “this is the first time I have this experience”? Don’t you think there has to be some kind of problems with Beauty’s knowledge state? The situation whre due to memory erasure the Beauty loses the ability to coherently reason about some statements makes much more sense than the alternative proposed by thirdism—according to which the Beauty becomes more confident in the state of the coin than she would’ve been if she didn’t have her memory erased.
Another intuition pump is that “today is Monday” is not actually True xor False from the perspective of the Beauty. From her perspective it’s True xor (True and False). This is because on Tails, the Beauty isn’t reasoning just for some one awakening—she is reasoning for both of them at the same time. When she awakens the first time the statement “today is Monday” is True, and when she awakens the second time the same statement is False. So the statement “today is Monday” doesn’t have stable truth value throughout the whole iteration of probability experiment. Suppose that Beauty really does not want to make false statements. Deciding to say outloud “Today is Monday”, leads to making a false statement in 100% of iterations of experiemnt when the coin is Tails.
P(today is Monday | heads) = 100% is fine. (Or is that tails? I keep forgetting.) P(today is Monday | tails) = 50% is fine too. (Or maybe it’s not? Maybe this is where I’m going working? Needs a bit of work but I suspect I could formalize that one if I had to.) But if those are both fine, we should be able to combine them, like so: heads and tails are mutually exclusive and one of them must happen, so: P(today is Monday) = P(heads) • P(today is Monday | heads) + P(tails) • P(today is Monday | tails) = 0.5 + .25 = 0.75 Okay, I was expecting to get 2⁄3 here. Odd. More to the point, this felt like cheating and I can’t put my finger on why. maybe need to think more later
Here you are describing Lewis’s model which is appropriate for Single Awakening Problem. There the Beauty is awakened on Monday if the coin is Heads, and if the coin is Tails, she is awakened either on Monday or on Tuesday (not both). It’s easy to see that 75% of awakening in such experiment indeed happen on Monday.
It’s very good that you notice this feeling of cheating. This is a very important virtue. This is what helped me construct the correct model and solve the problem in the first place—I couldn’t accept any other—they all were somewhat off.
I think, you feel this way, because you’ve started solving the problem from the wrong end, started arguing with math, instead of accepting it. You noticed that you can’t define “Today is Monday” normally so you just assumed as an axiom that you can.
But as soon as you assume that “Today is Monday” is a coherent event with a stable truth value throughout the experiment, you inevitably start talking about a different problem, where it’s indeed the case. Where there is only one awakening in any iteration of probability experiment and so you can formally construct a sample space where “Today is Monday” is an elementary mutually exclusive outcome. There is no way around it. Either you model the problem as it is, and then “Today is Monday” is not a coherent event, or you assume that it is coherent and then you are modelling some other problem.
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.
Exactly! I’m glad that you actually engaged with the problem.
The first step is to realize that here “today” can’t mean “Monday xor Tuesday” because such event never happens. On every iteration of experiment both Monday and Tuesday are realized. So we can’t say that the participant knows that they are awakened on Monday xor Tuesday.
Can we say that participant knows that they are awakened on Monday or Tuesday? Sure. As a matter of fact:
P(Monday or Tuesday) = 1
P(Heads|Monday or Tuesday) = P(Heads) = 1⁄2
This works, here probability that the coin is Heads in this iteration of the experiment happens to be the same as what our intuition is telling us P(Heads|Today) is supposed to be, however we still can’t define “Today is Monday”:
P(Monday|Monday or Tuesday) = P(Monday) = 1
Which doesn’t fit our intuition.
How can this be? How can we have a seeminglly well-defined probability for “Today the coin is Heads” but not for “Today is Monday”? Either “Today” is well-defined or it’s not, right? Take some time to think about it.
What do we actually mean when we say that on an awakening the participant supposed to believe that the coin is Heads with 50% probability? Is it really about this day in particular? Or is it about something else?
The answer is: we actually mean, that on any day of the experiment be it Monday or Tuesday the participant is supposed to believe that the coin is Heads with 50% probability. We can not formally specify “Today” in this problem but there is a clever, almost cheating way to specify “Anyday” without breaking anything.
This is not easy. It requires a way to define P(A|B), when P(B) itself is undefined which is unconventional. But, moreover, it requires symmetry. P(Heads|Monday) has to be equal to P(Heads|Tuesday) only then we have a coherent P(Heads|Anyday).
This makes me uncomfortable. From the perspective of sleeping beauty, who just woke up, the statement “today is Monday” is either true or false (she just doesn’t know which one). Yet you claim she can’t meaningfully assign it a probability. This feels wrong, and yet, if I try to claim that the probability is, say, 2⁄3, then you will ask me “in what sample space?” and I don’t know the answer.
What seems clear is that the sample space is not the usual sleeping beauty sample space; it has to run metaphorically “skew” to it somehow.
If the question were “did the coin land on heads” then it’s clear that this is question is of the form “what world am I in?”. Namely, “am I in a world where the coin landed on heads, or not?”
Likewise if we ask “does a Tuesday awakening happen?”… that maps easily to question about the coin, so it’s safe.
But there should be a way to ask about today as well, I think. Let’s try something naive first and see where it breaks. P(today is Monday | heads) = 100% is fine. (Or is that tails? I keep forgetting.) P(today is Monday | tails) = 50% is fine too. (Or maybe it’s not? Maybe this is where I’m going working? Needs a bit of work but I suspect I could formalize that one if I had to.) But if those are both fine, we should be able to combine them, like so: heads and tails are mutually exclusive and one of them must happen, so: P(today is Monday) = P(heads) • P(today is Monday | heads) + P(tails) • P(today is Monday | tails) = 0.5 + .25 = 0.75 Okay, I was expecting to get 2⁄3 here. Odd. More to the point, this felt like cheating and I can’t put my finger on why. maybe need to think more later
Where does the feeling of wrongness come from? Were you under impression that probability theory promised us to always assign some measure to any statement in natural language? It just so happens that most of the time we can construct an appropriate probability space. But the actual rule is about whether or not we can construct a probability space, not whether or not something is a statement in natural language.
Is it really so surprising that a person who is experiencing amnesia and the repetetion of the same experience, while being fully aware of the procedure can’t meaningfully assign credence to “this is the first time I have this experience”? Don’t you think there has to be some kind of problems with Beauty’s knowledge state? The situation whre due to memory erasure the Beauty loses the ability to coherently reason about some statements makes much more sense than the alternative proposed by thirdism—according to which the Beauty becomes more confident in the state of the coin than she would’ve been if she didn’t have her memory erased.
Another intuition pump is that “today is Monday” is not actually True xor False from the perspective of the Beauty. From her perspective it’s True xor (True and False). This is because on Tails, the Beauty isn’t reasoning just for some one awakening—she is reasoning for both of them at the same time. When she awakens the first time the statement “today is Monday” is True, and when she awakens the second time the same statement is False. So the statement “today is Monday” doesn’t have stable truth value throughout the whole iteration of probability experiment. Suppose that Beauty really does not want to make false statements. Deciding to say outloud “Today is Monday”, leads to making a false statement in 100% of iterations of experiemnt when the coin is Tails.
Here you are describing Lewis’s model which is appropriate for Single Awakening Problem. There the Beauty is awakened on Monday if the coin is Heads, and if the coin is Tails, she is awakened either on Monday or on Tuesday (not both). It’s easy to see that 75% of awakening in such experiment indeed happen on Monday.
It’s very good that you notice this feeling of cheating. This is a very important virtue. This is what helped me construct the correct model and solve the problem in the first place—I couldn’t accept any other—they all were somewhat off.
I think, you feel this way, because you’ve started solving the problem from the wrong end, started arguing with math, instead of accepting it. You noticed that you can’t define “Today is Monday” normally so you just assumed as an axiom that you can.
But as soon as you assume that “Today is Monday” is a coherent event with a stable truth value throughout the experiment, you inevitably start talking about a different problem, where it’s indeed the case. Where there is only one awakening in any iteration of probability experiment and so you can formally construct a sample space where “Today is Monday” is an elementary mutually exclusive outcome. There is no way around it. Either you model the problem as it is, and then “Today is Monday” is not a coherent event, or you assume that it is coherent and then you are modelling some other problem.
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 ;)