Both half and thirder arguments are correct and defensible. They are the correct answers to 2 different questions. I don’t think I have the skill to disambiguate the question, but seems clear to me it is ambiguous.
I think the argument for ambiguity can be stated more simply.
Suppose the sleeper is allowed to bet on H or T on each awakening, and is given $1 for each correct response (and $0 for each incorrect response). With this protocol, the sleeper should always bet T, and will accumulate a number of dollars that is 2/3 (on average) of the number of coin flips. This is the "thirder" position.
But now suppose the sleeper decides in advance to always bet T, and the payoff is that $1 is awarded for each coin flip that comes up T. Now the sleeper accumulate a number of dollars that is 1/2 (on average) of the number of coin flips. This is the "halfer" position.
So it just depends on which betting protocol embodies the meaning of "degree of belief", which is ambigious otherwise.
Reward structure shouldn't affect your probability estimate of the event.
Suppose a coin is tossed and you can bet on the outcome with the condition that if it's Tails the bet is repeated. Does it mean that the probability of a coin to land Heads is only 1/3?
In the terminology of your lesswrong piece, it means the weighted probability is 1/3.
Reward structure determines whether "degree of belief" should be (in your terminology) probability or weighted probability.
I would further argue that your "weighted probability" is simply Bayesian probability in a larger context where there is no sample space, and that we need not invent a new term for that, but that's mere semantics.
> In the terminology of your lesswrong piece, it means the weighted probability is 1/3.
This is correct. Probability is 1/2. Probability weighted by the number of bets is 1/3.
> Reward structure determines whether "degree of belief" should be (in your terminology) probability or weighted probability.
And this is wrong. If you want to claim that sometimes probability is not degree of belief you will have to disprove all the foundations of probability theory.
Notice that actually, both probability and weighted probability allow to arrive to correct betting odds, regardless of the reward structure:
> I would further argue that your "weighted probability" is simply Bayesian probability in a larger context where there is no sample space, and that we need not invent a new term for that, but that's mere semantics.
Oh, its not semantics at all!
Bayesian probability is probability. It corresponds to the same axioms including the necessity of sample space, that consists of mutually exclusive outcomes.
Weighted probability is not probability. It may look very similar but it has one less restriction which makes it a different entity.
Once again, as a sanity check, are you really claiming that Bayesian probability of a fair coin to land Heads can be anything depending on the reward structure? That probability is a function of utility? If so, how can you have degree of belief when you don't know the reward structure, or when there is no reward structure at all?
My bottom line is that I view the whole problem as a semantics issue with no deep significance.
Everybody agrees on what is happening physically. Everybody agrees on how to bet, given a fully specified reward scheme.
The ONLY disagreement is on how to define the phrase "degree of belief".
And I just don't think that is a very interesting question. You like the halfer definition. Great, knock yourself out trying to convince the thirders!
FWIW I was originally a halfer, but a long in-person argument with a thirder (for whom I had the highest intellectual respect) left me with my current view, that it is all just a matter of definition.
So imagine the following case: A spaceship leaves from Earth headed to Planet Z, and the inhabitants are put to sleep for the duration of the long journey. 1% of the population has Sleeping Beauty Syndrome (SBS), which has 2 effects: first, those with SBS will wake up from their sleep exactly 9,801 times over the course of the journey, while those without SBS awake only once on the journey. Second, the atmosphere of Planet Z is toxic to anyone with SBS but fine for those without SBS, so those with SBS must take a pill every time they wake up over the course of the journey to Planet Z or else they will die upon arrival. However, if a human WITHOUT SBS takes the pill, it will kill them instantly. No one knows whether they have SBS or not, and a pill is offered to every passenger when they wake.
Knowing all of this, you wake up aboard this spaceship with a pill sitting in front of you. Should you take it?
This is the "halfer" position in the original, and corresponds to a betting protocol of payoff per flip, not payoff per awakening (see my other comment).
Isn't that the "thirder" position? Wouldn't a halfer reason that waking up once doesn't give you any useful information for telling whether you have SBS or not, so you should take the pill because the odds that you have SBS are only 1%?
EDIT: Never mind, I misread the question and got it flipped, thought those with SBS will die if they take the pill.
I don't understand your argument, and must be missing something. In the original Sleeping Beauty scenario halving makes sense to me: no matter which way the coin flipped you will make the same observations, so your observations don't provide information on how the coin flip turned out. But in your modified scenario you will not observe the same things regardless of the coin flip, so what you observe does provide you with information. Namely, you can observe if you've been awake for 15 minutes or not. If you've been awake longer than 15 minutes then that's an observation that you would expect to observe in 3 out of the 4 waking up options, so it makes sense that observing you're not in the 4th possible option changes your probability of whether it was heads or tails.
I guess another way to put it is that in the original problem observing that I am awake and have no memory of waking before tells me nothing, because I'd observe that in all three possible waking ups. But observing I've been awake more than 15 minutes in your version tells me that it's either Monday or the coin came up heads. Shouldn't learning that it can't be Tuesday with tails change your probabilities in this case?
Just stipulate that the pill must be taken within 5 minutes, and re-sleep comes only after 15 minutes.
Clearly it's necessary to set up the scenario so that no new information is conveyed on each awakening, as in the original.
The real issue is whether rewards for repeated awakenings are cumumlative or not. If they are, thirders are correct; if they are not, halfers are correct.
In Dan's scenario, rewards are not cumulative; taking the pill multiple times is no better than taking it once.
It is a question about probability (more precisely in this case, "degree of belief", also called "credence"). But what does "degree of belief" mean in practice? A standard way of specifying this is via a betting (or rewards) protocol. For example: I am going to flip a coin that you believe has been weighted to come up H 1/3 of the time and T 2/3 of the time. I offer you a choice: I will pay you $1 if it comes up T, OR $X if it comes up H. What would X have to be for you to have no preference between those two choices? Given your degree of belief, $X would need to be $2. So by asking you what value of X makes the two rewards equal in your mind, I can deduce your degree of belief.
Regarding the Sleeping Beauty Problem, my claim is that the reason it is contentious is that the rewards protocol is ambigous. This is because the scenario includes forgetting something that just happened, which is not common in everyday life! So our intuition is thrown off.
In terms of a rewards protocol, the issue is whether the sleeper gets to keep all rewards at the end, or instead forgets about (and does not keep) a reward given just before being put back to sleep. The first is the thirder position, the second is the halfer position.
So it comes down to which rewards protocol is implied by the wording of the problem. Each side passionately believes in their own interpretation. But this is a semantics issue only. There is nothing deep or profound here. And this is why neither side ever wins the argument.
> I claim that this is relevantly like the original sleeping beauty problem.
This claim is wrong.
The signature feature of Sleeping Beauty problem is that the Beauty is unable to distinguish between Monday and Tuesday awakening, she receives the same information and produces the same reasoning, to the point where statement "Today is Monday" doesn't have a coherent truth value, according to Beauty's epistemic state: on Tails this statement is simultaneously True and False in the same iteration of probability experiment.
This is a very delicate state of affairs. Even a minor pertrubation breaks the symmetry. For example, provide the Beauty with any way to generate random numbers. Now she has a strategy to guess Tails with nearly 2/3 accuracy *per experiment*. All she needs to do is to precommit to a particular number n, generate a number on every awakening and guess Tails, only when n is generated, refusing to guess otherwise.
Compared to that, what you are doing is smashing the whole setting of the experiment with a sledgehammer. You are givng the Beauty new evidence in a very clear way! The fact that you do not notice a problem means that you've never really engaged with Halfing in Sleeping Beauty beyond noticing that obviously wrong model of David Lewis is indeed obviously wrong.
> First of all, the thing that matters isn’t actually if you performed the reasoning but what your credence should have been.
> Second, this gives bizarre results in the case where you’re awake for 15 minutes—it implies that if during the 15 minutes you didn’t think about anthropics, then after the 15 minutes pass, you should be at 1/2 on the coin coming up tails, rather than 2/3.
At this point you need to strictly specify what does "you" mean. Are we talking about idealized agents, capable of noticig even the quickest of events, or are we talking about rational reasoners who have human limitations, or are we talking about some brand of irrational human.
Usually the question is about the rational behavior of a human. Someone who can't notice being awoken for half a second but does notices being awoken for fifteen minutes. And its assumed that such agent does notice everything that they can. So we say that an awakening for 15 minutes is enough for update, while awakening for half a second is not.
If you want to talk about perfect entities without any human limitations than you can't appeal to human intuition that "surely the half-a-second of waking shouldn’t make a difference". If you want to talk about an irrational human who doesn't pay attention to being awaken for 15 minutes, then you can't complain that its a bizarre behavior.
If you are in general weirded out by the fact that mind state and reasoning capabilities of an agent affects what events they can observe, notice that people who are full-blind, color blind and have perfect vision naturally observe very different events.
> Upon waking up, both heads and tails were equally likely, so now that you’ve gotten evidence (remaining awake) that’s twice as strongly predicted on heads, you should be at 2/3 on heads.
Typo. You meant "twice as strongly predicted on *tails*, you should be at 2/3 on *tails*".
> Suppose you wake up and remain awake for 10 minutes. You discover a button that will, if pressed, make it so that if it’s the first day and the coin came up heads, you won’t awake the second day. On this view, because the second-day momentary waking makes a difference to how you should probabilistically reason, even though you know that’s a state you’re not in, your credence should be different before and after pressing the button.
Lets replace momentary second awakening on Heads with an awakening for full 5 minutes, to sceen off the previous concerns. Now what happens when you press the button before/after the 5 minutes have passed?
In any case you know that you've pressed it in your first awakening, which means that second awakening on Heads doesn't happen, therefore being awaken for 10 minutes doesn't reveal any information to you anymore, as you would've been awaken for 10 minutes anyway. I'm not sure why you find this situation weird. Everything is very straightforward: past actions affect the future.
Compare it with thirder Beauty being able to retroactively change her supposedly rational probability estimate during the experiment to any value she likes, by giving herself several extra awakenings, after the experiment has already ended, based on the revealed state of the coin. Now this is really weird.
To put it much more simply, the prior probability of the coin landing heads is 1/2 until you learn the rules of the game, at which point the prior probability you should assign switches to 1/3. This does not change subsequently, as no new information is introduced, so the thirder position is established once the rules are of the game are established, and remains so throughout.
If you're awake longer than 15 minutes then you know it's either Monday or Tuesday and the coin was heads, doesn't that give you some information on how likely it is to be Monday?
No, because you already knew that you would have the exact same mental state that you currently have, so having the mental state gievs you zero new information.
You knew that you would 100% experience a mental state where you did not fall asleep after 15 minutes (and you'll never have a mental state where you knew you fell asleep after 15 minutes, because you fall asleep instantly w/o notice)
Both half and thirder arguments are correct and defensible. They are the correct answers to 2 different questions. I don’t think I have the skill to disambiguate the question, but seems clear to me it is ambiguous.
I think this is basically right - Ray Briggs has, I think, the best paper on Sleeping Beauty, and argues essentially this position with more details. Briggs's paper can be found here: https://joelvelasco.net/teaching/3865/briggs10-puttingavalueonbeauty.pdf
I think the argument for ambiguity can be stated more simply.
Suppose the sleeper is allowed to bet on H or T on each awakening, and is given $1 for each correct response (and $0 for each incorrect response). With this protocol, the sleeper should always bet T, and will accumulate a number of dollars that is 2/3 (on average) of the number of coin flips. This is the "thirder" position.
But now suppose the sleeper decides in advance to always bet T, and the payoff is that $1 is awarded for each coin flip that comes up T. Now the sleeper accumulate a number of dollars that is 1/2 (on average) of the number of coin flips. This is the "halfer" position.
So it just depends on which betting protocol embodies the meaning of "degree of belief", which is ambigious otherwise.
Reward structure shouldn't affect your probability estimate of the event.
Suppose a coin is tossed and you can bet on the outcome with the condition that if it's Tails the bet is repeated. Does it mean that the probability of a coin to land Heads is only 1/3?
In the terminology of your lesswrong piece, it means the weighted probability is 1/3.
Reward structure determines whether "degree of belief" should be (in your terminology) probability or weighted probability.
I would further argue that your "weighted probability" is simply Bayesian probability in a larger context where there is no sample space, and that we need not invent a new term for that, but that's mere semantics.
> In the terminology of your lesswrong piece, it means the weighted probability is 1/3.
This is correct. Probability is 1/2. Probability weighted by the number of bets is 1/3.
> Reward structure determines whether "degree of belief" should be (in your terminology) probability or weighted probability.
And this is wrong. If you want to claim that sometimes probability is not degree of belief you will have to disprove all the foundations of probability theory.
Notice that actually, both probability and weighted probability allow to arrive to correct betting odds, regardless of the reward structure:
https://www.lesswrong.com/posts/cvCQgFFmELuyord7a/beauty-and-the-bets
> I would further argue that your "weighted probability" is simply Bayesian probability in a larger context where there is no sample space, and that we need not invent a new term for that, but that's mere semantics.
Oh, its not semantics at all!
Bayesian probability is probability. It corresponds to the same axioms including the necessity of sample space, that consists of mutually exclusive outcomes.
Weighted probability is not probability. It may look very similar but it has one less restriction which makes it a different entity.
Once again, as a sanity check, are you really claiming that Bayesian probability of a fair coin to land Heads can be anything depending on the reward structure? That probability is a function of utility? If so, how can you have degree of belief when you don't know the reward structure, or when there is no reward structure at all?
My bottom line is that I view the whole problem as a semantics issue with no deep significance.
Everybody agrees on what is happening physically. Everybody agrees on how to bet, given a fully specified reward scheme.
The ONLY disagreement is on how to define the phrase "degree of belief".
And I just don't think that is a very interesting question. You like the halfer definition. Great, knock yourself out trying to convince the thirders!
FWIW I was originally a halfer, but a long in-person argument with a thirder (for whom I had the highest intellectual respect) left me with my current view, that it is all just a matter of definition.
Only one of the questions is about mathematical notion of probability, though.
I'm disentangling the semantic disagreement here: https://www.lesswrong.com/posts/Gf4WtPfrELwRtfaM9/semantic-disagreement-of-sleeping-beauty-problem
Feel free to read the previous posts as well, if you are curious about the actual disagreement between halfers and thirders.
Woah, we agree!
So imagine the following case: A spaceship leaves from Earth headed to Planet Z, and the inhabitants are put to sleep for the duration of the long journey. 1% of the population has Sleeping Beauty Syndrome (SBS), which has 2 effects: first, those with SBS will wake up from their sleep exactly 9,801 times over the course of the journey, while those without SBS awake only once on the journey. Second, the atmosphere of Planet Z is toxic to anyone with SBS but fine for those without SBS, so those with SBS must take a pill every time they wake up over the course of the journey to Planet Z or else they will die upon arrival. However, if a human WITHOUT SBS takes the pill, it will kill them instantly. No one knows whether they have SBS or not, and a pill is offered to every passenger when they wake.
Knowing all of this, you wake up aboard this spaceship with a pill sitting in front of you. Should you take it?
Absolutely not!
This is the "halfer" position in the original, and corresponds to a betting protocol of payoff per flip, not payoff per awakening (see my other comment).
Agreed!
Isn't that the "thirder" position? Wouldn't a halfer reason that waking up once doesn't give you any useful information for telling whether you have SBS or not, so you should take the pill because the odds that you have SBS are only 1%?
EDIT: Never mind, I misread the question and got it flipped, thought those with SBS will die if they take the pill.
I don't understand your argument, and must be missing something. In the original Sleeping Beauty scenario halving makes sense to me: no matter which way the coin flipped you will make the same observations, so your observations don't provide information on how the coin flip turned out. But in your modified scenario you will not observe the same things regardless of the coin flip, so what you observe does provide you with information. Namely, you can observe if you've been awake for 15 minutes or not. If you've been awake longer than 15 minutes then that's an observation that you would expect to observe in 3 out of the 4 waking up options, so it makes sense that observing you're not in the 4th possible option changes your probability of whether it was heads or tails.
I guess another way to put it is that in the original problem observing that I am awake and have no memory of waking before tells me nothing, because I'd observe that in all three possible waking ups. But observing I've been awake more than 15 minutes in your version tells me that it's either Monday or the coin came up heads. Shouldn't learning that it can't be Tuesday with tails change your probabilities in this case?
Just stipulate that the pill must be taken within 5 minutes, and re-sleep comes only after 15 minutes.
Clearly it's necessary to set up the scenario so that no new information is conveyed on each awakening, as in the original.
The real issue is whether rewards for repeated awakenings are cumumlative or not. If they are, thirders are correct; if they are not, halfers are correct.
In Dan's scenario, rewards are not cumulative; taking the pill multiple times is no better than taking it once.
Was this meant to be a reply to me? If it was, then I'm sorry but I'm still confused.
It was a reply to you, and I've explained it as best I can.
What do rewards have to do with the Sleeping Beauty problem? I thought it was a question about probability.
It is a question about probability (more precisely in this case, "degree of belief", also called "credence"). But what does "degree of belief" mean in practice? A standard way of specifying this is via a betting (or rewards) protocol. For example: I am going to flip a coin that you believe has been weighted to come up H 1/3 of the time and T 2/3 of the time. I offer you a choice: I will pay you $1 if it comes up T, OR $X if it comes up H. What would X have to be for you to have no preference between those two choices? Given your degree of belief, $X would need to be $2. So by asking you what value of X makes the two rewards equal in your mind, I can deduce your degree of belief.
Regarding the Sleeping Beauty Problem, my claim is that the reason it is contentious is that the rewards protocol is ambigous. This is because the scenario includes forgetting something that just happened, which is not common in everyday life! So our intuition is thrown off.
In terms of a rewards protocol, the issue is whether the sleeper gets to keep all rewards at the end, or instead forgets about (and does not keep) a reward given just before being put back to sleep. The first is the thirder position, the second is the halfer position.
So it comes down to which rewards protocol is implied by the wording of the problem. Each side passionately believes in their own interpretation. But this is a semantics issue only. There is nothing deep or profound here. And this is why neither side ever wins the argument.
> I claim that this is relevantly like the original sleeping beauty problem.
This claim is wrong.
The signature feature of Sleeping Beauty problem is that the Beauty is unable to distinguish between Monday and Tuesday awakening, she receives the same information and produces the same reasoning, to the point where statement "Today is Monday" doesn't have a coherent truth value, according to Beauty's epistemic state: on Tails this statement is simultaneously True and False in the same iteration of probability experiment.
This is a very delicate state of affairs. Even a minor pertrubation breaks the symmetry. For example, provide the Beauty with any way to generate random numbers. Now she has a strategy to guess Tails with nearly 2/3 accuracy *per experiment*. All she needs to do is to precommit to a particular number n, generate a number on every awakening and guess Tails, only when n is generated, refusing to guess otherwise.
Compared to that, what you are doing is smashing the whole setting of the experiment with a sledgehammer. You are givng the Beauty new evidence in a very clear way! The fact that you do not notice a problem means that you've never really engaged with Halfing in Sleeping Beauty beyond noticing that obviously wrong model of David Lewis is indeed obviously wrong.
> First of all, the thing that matters isn’t actually if you performed the reasoning but what your credence should have been.
> Second, this gives bizarre results in the case where you’re awake for 15 minutes—it implies that if during the 15 minutes you didn’t think about anthropics, then after the 15 minutes pass, you should be at 1/2 on the coin coming up tails, rather than 2/3.
At this point you need to strictly specify what does "you" mean. Are we talking about idealized agents, capable of noticig even the quickest of events, or are we talking about rational reasoners who have human limitations, or are we talking about some brand of irrational human.
Usually the question is about the rational behavior of a human. Someone who can't notice being awoken for half a second but does notices being awoken for fifteen minutes. And its assumed that such agent does notice everything that they can. So we say that an awakening for 15 minutes is enough for update, while awakening for half a second is not.
If you want to talk about perfect entities without any human limitations than you can't appeal to human intuition that "surely the half-a-second of waking shouldn’t make a difference". If you want to talk about an irrational human who doesn't pay attention to being awaken for 15 minutes, then you can't complain that its a bizarre behavior.
If you are in general weirded out by the fact that mind state and reasoning capabilities of an agent affects what events they can observe, notice that people who are full-blind, color blind and have perfect vision naturally observe very different events.
> Upon waking up, both heads and tails were equally likely, so now that you’ve gotten evidence (remaining awake) that’s twice as strongly predicted on heads, you should be at 2/3 on heads.
Typo. You meant "twice as strongly predicted on *tails*, you should be at 2/3 on *tails*".
> Suppose you wake up and remain awake for 10 minutes. You discover a button that will, if pressed, make it so that if it’s the first day and the coin came up heads, you won’t awake the second day. On this view, because the second-day momentary waking makes a difference to how you should probabilistically reason, even though you know that’s a state you’re not in, your credence should be different before and after pressing the button.
Lets replace momentary second awakening on Heads with an awakening for full 5 minutes, to sceen off the previous concerns. Now what happens when you press the button before/after the 5 minutes have passed?
In any case you know that you've pressed it in your first awakening, which means that second awakening on Heads doesn't happen, therefore being awaken for 10 minutes doesn't reveal any information to you anymore, as you would've been awaken for 10 minutes anyway. I'm not sure why you find this situation weird. Everything is very straightforward: past actions affect the future.
Compare it with thirder Beauty being able to retroactively change her supposedly rational probability estimate during the experiment to any value she likes, by giving herself several extra awakenings, after the experiment has already ended, based on the revealed state of the coin. Now this is really weird.
To put it much more simply, the prior probability of the coin landing heads is 1/2 until you learn the rules of the game, at which point the prior probability you should assign switches to 1/3. This does not change subsequently, as no new information is introduced, so the thirder position is established once the rules are of the game are established, and remains so throughout.
I have published an article on here very recently about this. https://leightonvaughanwilliams.substack.com/p/the-sleeping-beauty-problem
Why isn't this just 15 minutes (generously) of writing code in one's language of choice to work out the probability? You could solve this in Excel.
Whatever you'd write the code for, it would be philosophically contentious whether that represents rational credences.
> if it came up heads, there’s a 1/2 chance you’d be asleep by now, while if it came up tails, you’d definitely still be awake now.
But your chances of having the mental state of being awake after 15 minutes is 100%, so it predicts nothing.
It's true that both theories predict you'd have that mental state at some point, but one makes it likelier you'd have it today!
"today" is a meaningless concept to someone who, by stipulation, is totally divorced from any information about what day it is.
If you're awake longer than 15 minutes then you know it's either Monday or Tuesday and the coin was heads, doesn't that give you some information on how likely it is to be Monday?
No, because you already knew that you would have the exact same mental state that you currently have, so having the mental state gievs you zero new information.
But I didn't know that: I knew that there was a 25% chance I'd fall asleep after 15 minutes. That's a different mental state!
You knew that you would 100% experience a mental state where you did not fall asleep after 15 minutes (and you'll never have a mental state where you knew you fell asleep after 15 minutes, because you fall asleep instantly w/o notice)