The Sleeping Beauty Problem: Any halfers here?

[stevendaryl]

The controversy is over whether Sleeping Beauty has any information that would allow (force?) her to update her estimate, from 1/2 before the experiment to 2/3 upon being awakened.

The weird thing is that a slightly modified experiment has a clear 2/3 answer, and it seems to me that there is no significant difference with the original version. Instead of wiping Sleeping Beauty's memory only when heads is thrown, you wipe her memory regardless, but in the case of tails, you restore her memory an hour after waking. Now, when she first wakes up (before the possibility of any memory restoration), you ask her what the likelihood of the four possibilities:

1. Heads and today is Monday
2. Heads and today is Tuesday
3. Tails and today is Monday
4. Tails and today is Tuesday

I think halfers would agree that all 4 possibilities are equally likely. An hour later, you again ask her the probabilities. If she did not have a memory restoration, then she knows that possibility 4 is ruled out. So her updated probabilities in light of that information is that the first 3 are all equally likely (as they were before), but the last is impossible.

So I would say that it isn't an assumption that possibilities 1-3 are equally likely. It's a conclusion from the fact that 1-4 were equally likely, and then 4 was eliminated as a possibility.

 

[Marana]

There is no reason not to use probability theory in this case.

We all agree that without the amnesia drug you wouldn't use relative frequency of tails awakenings to decide your subjective credence. Without the drug 1/3 of awakenings are heads. An outside observer should believe 1/3, because for them the awakening is a random experiment. But everyone, including you, agrees that sleeping beauty does not have subjective credence of 1/3. Now, we add the drug. But waking up is still not a random experiment from the perspective of sleeping beauty; modeling it like that does not reflect her information. It is unexplained why the drug means relative frequency of awakenings (as opposed to relative frequency of coin flips or something else entirely) is the only way one might define subjective credence that the coin flipped heads.

And I think we can agree that there are situations where you can't condition on things like "it is monday". For example, suppose you flip a coin while the clock says 2:00. "It is 2:00". While you compute the probability of heads you notice that the clock now says 2:01. "It is 2:01". Using conditioning, you notice that this is impossible (in conditioning something you know can't become false, nor can something you know is false become true) and give up. Or more likely, you don't give up because you intuitively know that you can't simply condition as normal with "it is 2:00" and "it is 2:01".

I suggest that if you have a correspondence of information between a starting time and a specified time, and the only difference is the passage of time, then if you learn that you in fact are at the specified time your probabilities in anything that doesn't change over time should be the same as at the starting point.

 

[Boing3000]

What is Sleeping Beauty's credence now for the proposition that the coin landed heads?

I don't even understand the problem.
The fair coin if flipped once. So 1/2 is the only answer possible.

In the wiki article the babbling about day's name and persuasion ("and comes to fully believe that it is Monday") is definitely the source of the confusion (hence my vote)
That's the probability to be Tuesday that is 1/3.

 

[Charles Link]

Had my example, for a biased coin, in post #99, come up with the same type of consistent algebraic calculation for the probability that seemed reasonable, I would have been convinced that the correct answer is 1/3. Instead though, for the biased coin, the choices for Sleeping Beauty are quite simple: 1)whether the unlikely (tails) occurred and she is on a long string of the unlikely, or 2) that the likely occurred (and it came up heads). $\\$ I am starting to wonder whether this question needs another choice=similar to answer #3, that explains it as a dilemma that defies logic and/or probability theory as we know it. Essentially, the coin is only flipped once, so we really aren't justified in doing a statistical sampling with many trials.

 

[stevendaryl]

And I think we can agree that there are situations where you can't condition on things like "it is monday". For example, suppose you flip a coin while the clock says 2:00. "It is 2:00". While you compute the probability of heads you notice that the clock now says 2:01. "It is 2:01". Using conditioning, you notice that this is impossible (in conditioning something you know can't become false, nor can something you know is false become true) and give up. Or more likely, you don't give up because you intuitively know that you can't simply condition as normal with "it is 2:00" and "it is 2:01".

That's an interesting problem--how to make sense of time-dependent statements such as "It is now 2:00". Later finding out that it is 2:01 doesn't contradict the truth of the earlier statement. However, I think you're barking up the wrong tree in relating it to the Sleeping Beauty problem--that's not really what's at issue.

 

[PeroK]

I don't even understand the problem.
The fair coin if flipped once. So 1/2 is the only answer possible.

Even after you look at the coin?

What about if someone else looks at the coin and does something that gives you a hint that it's tails?

 

[PeroK]

As my arguments are now spread over 8 pages, here is my summary of the thirder case:

First, a definition of "credence", which I've borrowed from Wikipedia:

Credence is a statistical term that expresses how much a person believes that a proposition is true.[1] As an example, a reasonable person will believe with 50% credence that a fair coin will land on heads the next time it is flipped. If the prize for correctly predicting the coin flip is $100, then a reasonable person will wager $49 on heads, but they will not wager $51 on heads.

This, to me, means the probability that the person can calculate given all the information at their disposal. I.e. a "reasonable" person will not choose to ignore relevant information - for fear of losing a bet, let's say.

A) First, a simple problem from probability theory to set the scene.

You have three people in separate rooms. You toss a coin. If it's heads you talk to one person. If it's tails you talk to two of them. When you talk to one of them, they must give their credence that the coin is a Head. All the participants know this procedure.

From elementary probability theory, we know that the answer is 1/3.

There are, however, many people (including those on this thread) who would disagree with this and give the answer 1/2.

Note: if there is genuine doubt about this question, then there would be no need for the Sleeping Beauty question, with its amnesia drug. You could have all the arguments over this simpler question about whether 1/3 or 1/2 or "it depends" is the correct answer.

In this case, the person interviewed could give their credence as the pre-interview probability of 1/2. But, that ignores the relevant information that they have been interviewed, and is, therefore, by definition "unreasonable" - and it implies they will unnecessarily lose bets on this credence.

B) The Sleeping Beauty problem

When she is woken she has the original information about the problem and she now also has the information that she has been awakened and that she does not know what day of the week it is. It could be Monday or it could be Tuesday. And, using the relative frequencies of Monday and Tuesday awakenings in the experiment, she must conclude that:

The probability that it is Monday is 2/3 and Tuesday 1/3.

With this information she can calculate that

The probability that the coin was Heads is 1/3 and that it was Tails is 2/3.

She could choose to ignore these calculations. But, again, by the definition of "credence" it is unreasonable to ignore information you have or to refuse to perform calculations on that data - for fear of losing a bet.

If she chooses simply to use the pre-awakening probability of 1/2, then she is choosing not to use the post-awakening information, which is unreasonable and therefore, by definition, 1/2 cannot be her credence.

C) The random observer

Further justification of the sleeper's position can be given by assuming that a random observer happens on the experiment. For this we really need to assume that the experiment has "day 1" and "day 2", rather than Monday-Tuesday. In any case, we assume the random observer does not know what stage the experiment has reached, but otherwise knows the procedure.

This observer will have precisely the same information as the sleeper. The only difference is that if it is Tuesday, the sleeper will have experienced and forgotten day 1 (Monday) but the observer will simply not know whether it ever took place. This is, however, the same information.

From elemenary probability theory (the observer has not been given any drug, so there ought to be no arguments about his calculations), the random observer can calculate the same probabilities as above, with probability 1/3 that it is Heads.

D) The case for 1/2.

The original case for 1/2 was that the sleeper had "no new information" when she was wakened. And this creates something of a paradox.

An analysis of the information that the sleeper has, however, shows that this information has changed since the start of the experiment. In particular, the knowldege that it could be Tuesday and that she could have been awakened on Monday and forgotten it.

This original argument, therefore, fails.

If the sleeper simply interprets the question as the pre-awakening probability, then (as above) she is ignoring the relevant information that she is now awake after a sleep and that is therefore not her credence.

Moreover, if the credence of heads is 1/2, then it is easily shown that it must be Monday. Let $p$ be the probability that it is Monday:

$p(H) = p/2$

Hence $p = 1$

The answer of 1/2 for Heads (given that she is now awake) implies that it must be Monday. And, therefore, cannot be a valid answer given that she is now awake.

E) A precise statement of the answer.

In answer to the question: what is (or was) your credence that the coin is Heads at the start of the experiment? The answer is 1/2.

In answer to the question: what is (or will be) your credence that the coin is Heads after you have been woken? The answer is 1/3.

In terms of making the original problem precise, the only change I would make is to emphasise that the required credence is when the sleeper is awake (and not her credence at the start of the experiment).

That concludes my case for the thirder.

F) The remaining arguments against the thirder position are that the use of the relative frequencies is invalid in this case. But, for me, relative frequency is equivalent to probability. If probability theory is invalid in this case and cannot be used to get an answer of 1/3, then I fail to see how probability theory can be used to get an answer of 1/2.

We may conclude that the problem itself (on account of the amnesia drug) is not self-consistent. But, given the consistency of the sleeper's calculations with those of a random observer, I see no reason to place this problem outside the realm of probability theory.

 

[Boing3000]

Credence is a statistical term that expresses how much a person believes that a proposition is true.[1] As an example, a reasonable person will believe with 50% credence that a fair coin will land on heads the next time it is flipped. If the prize for correctly predicting the coin flip is $100, then a reasonable person will wager $49 on heads, but they will not wager $51 on heads.

A reasonable person don't wager.
Look, for once, I'll try to be dead serious. This problem is in the "positive trolling" category, exactly like the chicken and eggs. Those problems have perfectly fine and simple answer, until you add some "hidden" confusion based on tricky language where everyone feels untitled to plug their own meaning.
In sleeping beauty, the trolling start when you speak about days name. This is irrelevant because of the drug and the closed room.

So if the coin is fair, and the experiment is fair, Beauty does not even have to be put to sleep to answer the question. Actually the coins does not even have to be flipped before Tuesday. So asking the question on Monday is definitely some sort of lying if the experimenter choose to do so.

The probability that it is Monday is 2/3 and Tuesday 1/3.

I agree, but then it is irrelevant to the question "What is your credence now for the proposition that the coin landed heads?".

With this information she can calculate that
The probability that the coin was Heads is 1/3 and that it was Tails is 2/3.

I have no idea how Beauty or anyone else could do that calculation. "Waking up" have never changed anything to how you process the world, except maybe on the February 2 (arghh I lied, I am not serious anymore)

She could choose to ignore these calculations. But, again, by the definition of "credence" it is unreasonable to ignore information you have or to refuse to perform calculations on that data - for fear of losing a bet.

If the question was "how many time do you wake up", I would agree Beauty should answer 3/2.
But that is not the question, isn't it ?

 

[Marana]

Well, that's the step where probability changes from frequency (which is just a matter of counting) to likelihood (which is a measure of confidence in the truth of something). To apply probability in our own lives, we are always in unique situations. But if you imagine that your situation is one instance of an ensemble of similar situations, then you can reason about likelihood. If you don't take such a step, then I don't see how probability is either useful or meaningful.

To be more precise, I'm not arguing against considering oneself a random selection, nor am I arguing against using relative frequency. They just can't be applied to a single wake-up, which is neither an experiment at all, nor an adequate model of sleeping beauty's full situation.

Sleeping beauty can consider herself to be randomly selected from all such sleeping beauties. That would give a probability of 1/2. The question is whether, from that starting point of 1/2, there is any way to update to 1/3. That's what I haven't seen so far.

Equivalently, she can consider her experiment (the coin toss followed by either M or MT) to be randomly selected from all such experiments, and use the relative frequency of 1/2. This is justified because the experiment is an experiment. Again, the question is whether she can change from that 1/2.

Relative frequency can be used, but not haphazardly. It needs to be justified... and we all agree on that! Because clearly using relative frequency of heads awakenings is the wrong way to compute the subjective probability of a heads coin toss without the drug. That is a misuse. And I'm arguing that, even with the drug, it is still a misuse to use relative frequency of heads awakenings as the subjective probability of a heads coin toss. But it is a proper use to use relative frequency with experiments, since they are experiments.

That's an interesting problem--how to make sense of time-dependent statements such as "It is now 2:00". Later finding out that it is 2:01 doesn't contradict the truth of the earlier statement. However, I think you're barking up the wrong tree in relating it to the Sleeping Beauty problem--that's not really what's at issue.

It is at issue in the most popular argument for 1/3. The argument goes: after conditioning on "it is monday", we must get 1/2. That is, P(monday and heads)/P(monday) = P(monday and tails)/P(monday) = 1/2. Therefore P(monday and heads) = P(monday and tails).

I assume we all agree that you wouldn't use regular conditioning on "it is now 2:00" and "it is now 2:01" in my example. I argue that the same holds here. You can't use regular conditioning on "it is monday", and the argument for 1/3 doesn't work. This isn't really a new thing: none of us condition on the flow of time and immediately throw out every probability computation. It's unspoken, but we avoid it intuitively. Until the sleeping beauty problem, a bizarre setup that makes us want to use it.

Rather than using conditioning, what we usually do (correctly) is to know that, by itself, the passage of time doesn't change things that don't change with time. The probability of the coin is still 1/2 because the clock hands moving didn't change the probability of the coin. Not because I conditioned on the current time.

Now, if you did really want to condition on the time, you could treat the time as if it was randomly selected. That's fine. Then you would be conditioning on "it is time X and time X was randomly selected". Using this method you would get P(heads) = 1/2 and P(heads|monday) = 2/3. The typical halfer solution. It makes sense for P(heads|monday) = 2/3 because monday is more likely to be randomly selected on heads. I disagree with this model because I feel it isn't the best way to define the probability, but I don't disagree with the technique.

But thirders are trying to have it both ways. They do not want to treat time as randomly selected or static, yet they want to condition on it. They are using the "it is now 2:00" and "it is now 2:01" technique, and the only outcome of being a thirder (at least using this argument) is throwing out every probability calculation you've ever made. I disagree with both the model and the technique.

 

[PeroK]

A reasonable person don't wager.
Look, for once, I'll try to be dead serious. This problem is in the "positive trolling" category, exactly like the chicken and eggs. Those problems have perfectly fine and simple answer, until you add some "hidden" confusion based on tricky language where everyone feels untitled to plug their own meaning.
In sleeping beauty, the trolling start when you speak about days name. This is irrelevant because of the drug and the closed room.

So if the coin is fair, and the experiment is fair, Beauty does not even have to be put to sleep to answer the question. Actually the coins does not even have to be flipped before Tuesday. So asking the question on Monday is definitely some sort of lying if the experimenter choose to do so.I agree, but then it is irrelevant to the question "What is your credence now for the proposition that the coin landed heads?".I have no idea how Beauty or anyone else could do that calculation. "Waking up" have never changed anything to how you process the world, except maybe on the February 2 (arghh I lied, I am not serious anymore)If the question was "how many time do you wake up", I would agree Beauty should answer 3/2.
But that is not the question, isn't it ?

If you know how to write a computer programme, you could do a computer simulation of the problem and you would, to your surprise, see the answer 1/3 appear rather than 1/2.

If I go into computer mode. Simulation according to thirders:

Toss 1: H

Monday; wake Beauty: 1-0 to Heads

Toss 2: H

Monday wake Beauty: 2-0 Heads

Toss 3: T

Monday wake Beauty: 2-1 Heads
Tuesday wake Beauty: 2-2

Toss 4: H

Monday; wake Beauty: 3-2 to Heads

Toss 5: T

Monday wake Beauty: 3-3
Tuesday wake Beauty: 4-3 Tails

Toss 6: T

Monday wake Beauty: 5-3
Tuesday wake Beauty: 6-3 Tails

So, in that little simulation, we ended up with 3 Heads and 3 Tails for the coin. Beauty was woken 9 times (6 times on a Monday and 3 on a Tuesday). When she was woken, it was as a result of Heads 3 times and Tails 6 times.

Therefore, if she counted each time she was woken whether it had been a Head or a Tail, she would have got 6-3 tails. And, that is what's called a relative frequency and is the basis of what a probability is.

Now, the halfers position:

Simulation according to halfer's:

Toss 1: H

Monday; wake Beauty: 1-0 to Heads

Toss 2: H

Monday wake Beauty: 2-0 Heads

Toss 3: T

Monday wake Beauty: 2-1 Heads
Tuesday wake Beauty: (dummy bet, does not count)

Toss 4: H

Monday; wake Beauty: 3-1 to Heads

Toss 5: T

Monday wake Beauty: 3-2 Heads
Tuesday wake Beauty: (dummy bet, does not count)

Toss 6: T

Monday wake Beauty: 3-3
Tuesday wake Beauty: (dummy bet, does not count)

So, again we have 3 Heads and 3 Tails and Beauty is woken 9 times (with again 6 on a Monday). But, according to the halfers, the Tuesday wakenings don't count (if they were bets, the bets would be null and void) - because she has alraedy bet once on the same coin. So, by discounting Tuesday you get as many Heads as Tails.

So, halfers don't allow two bets on the same coin, but thirders do.

Finally, the thirders position is: if you don't allow the Tuesday bets to stand, why have them in the first place? Isn't that then a different problem?

Alternatively, the halfers don't accept that you can simulate the problem in this way, using relative frequencies. You only have one experiment, so it's pointless to think about what happens if you run the experiment many times.

Or, to be fair, the halfers position is that the experimenters can count relative frequencies, but not Beauty. She's not allowed to imagine what would happen if the experiment were repeated many times - or, at least, it's wrong for her to do it.

 

[PeroK]

For a bit of fun, I imagined @Marana in the role of experimenter and me in the role of sleeping beauty:

Me: You woke me up!
Marana: Yes, I need to ask you what is your credence that the coin is heads.

Me: What day is it?
Marana: I can't tell you that.

Me: Is it Monday?
Marana: I can't tell you that.

Me: Did you give me that drug already?
Marana: I can't tell you that either.

Me: I believe there is likelihood of 1/3 that it's heads.
Marana: That's wrong!

Me: Why is it wrong?
Marana: How did you get 1/3?

Me: I imagined that we did this experiment many times and ... therefore, by using relative frequencies I got 1/3.
Marana: But, your not allowed to think like that.

Me: How can you stop me thinking?
Marana: Okay, I can't stop you thinking, but it's wrong for you to think like that.

Me: Why is it wrong?
Marana: Because relative frequencies are invalid for you.

Me: Would you be allowed to use relative frequencies. Perhaps to explain this experiment to an observer?
Marana: Yes, of course.

Me: So, it's right for you to think like that but it's wrong for me to think like that?
Marana: Yes.

Me: But, I'm quite good at probabilities. Why can't I use my natural abilities here and now?
Marana: Because you can't.

Me: So, basically, I'm expected to give the dumb answer of 1/2 (because, hey, nothing can change a coin after it's been tossed) and not the clever answer of 1/3?
Morana: I'm afraid so.

Me: That's not fair!
Marana: That's the way it is.

 

[forcefield]

The coin flip creates two equally likely scenarios. Hence the probability is 1/2.

I was only a part-time halfer: I should add that she has information about the relative probabilities of her being awake in the two scenarios. Hence 1/3.

I was only part-time thirder: That relative probability is actually 1:1. The fact that SB is awake twice in case of tail and only once in case of head does not increase belief in tail. So I finally voted for 1/2.

 

[stevendaryl]

Sleeping beauty can consider herself to be randomly selected from all such sleeping beauties. That would give a probability of 1/2. The question is whether, from that starting point of 1/2, there is any way to update to 1/3. That's what I haven't seen so far.

I went through the numbers, and it's 2/3. If you do the experiment over and over, with different starting times, then of the "active" Sleeping Beauties (the ones in day 1 or day 2 of the experiment), 1/3 are on day 1 after a coin flip of heads, 1/3 are on day 2 after a coin flip of heads, and 1/3 are on day 1 after a coin flip of tails. If the actual Sleeping Beauty thinks of herself as a random choice among those, then she would come up with 2/3 heads.

It's sort of similar to the situation where there is some country where people have a 50% chance of producing one offspring, and 50% chance of producing two offspring. If you take a random adult and ask the probability that it will have two offspring, the answer is 50%. If you take a random child and ask what is the probability that their parent had two children, it's 2/3.

Equivalently, she can consider her experiment (the coin toss followed by either M or MT) to be randomly selected from all such experiments, and use the relative frequency of 1/2. This is justified because the experiment is an experiment. Again, the question is whether she can change from that 1/2.

Yeah, in the repeated sleeping beauty experiment, there are two different relative frequencies to compute:

• Let's call a coin flip "active" if it's being used in a sleeping beauty experiment, and it was flipped less than two days ago (regardless of the result).
• Let's call a sleeping beauty "active" if she is experiencing day 1 or day 2.

Then:

• 2/3 of active sleeping beauties have an associated coin toss result of heads
• 1/2 of all active coin flips are heads

So, if you point to a random coin flip result (recorded on a piece of paper) and ask a Sleeping Beauty what are the odds that it is heads, she should answer 1/2. But if you tell her that it's her result, she should say 2/3.

Relative frequency can be used, but not haphazardly. It needs to be justified... and we all agree on that! Because clearly using relative frequency of heads awakenings is the wrong way to compute the subjective probability of a heads coin toss without the drug. That is a misuse. And I'm arguing that, even with the drug, it is still a misuse to use relative frequency of heads awakenings as the subjective probability of a heads coin toss. But it is a proper use to use relative frequency with experiments, since they are experiments.

To call it a misuse, you need to say what reason is there not to. What harm comes from it?

To me, the best example of a counter-intuitive result coming from the thirder position is to change it to a lottery. A person has a one in a million chance of winning. But you can make it subjectively 50/50 by waking the winner a million days in a row. That's strange.

It is at issue in the most popular argument for 1/3. The argument goes: after conditioning on "it is monday", we must get 1/2.
That is, P(monday and heads)/P(monday) = P(monday and tails)/P(monday) = 1/2. Therefore P(monday and heads) = P(monday and tails).

I assume we all agree that you wouldn't use regular conditioning on "it is now 2:00" and "it is now 2:01" in my example.

That's not the same thing. You can eliminate that problem by making statements about connections between events. "The first time I looked at the clock, it was 2:00." "The second time I looked at the clock, it was 2:01". No contradiction.

I argue that the same holds here. You can't use regular conditioning on "it is monday", and the argument for 1/3 doesn't work.

I think that's barking up the wrong tree. Instead of Monday, think about Tuesday. You ask Sleeping Beauty what the probability of getting heads was. Now you tell her that today is Tuesday. Then she knows that she got heads. Of course, telling her "Today is Tuesday" gave her new information, and it allowed her to update her likelihood estimates. So if you have a theory of subjective probability that can't accommodate such an update, then I would say something is wrong with that theory.

This isn't really a new thing: none of us condition on the flow of time and immediately throw out every probability computation. It's unspoken, but we avoid it intuitively. Until the sleeping beauty problem, a bizarre setup that makes us want to use it.

I don't think that there is anything wrong with conditioning on the flow of time. If it's possible to forget what day it is, then knowing what day it is is additional information, and it can change your subjective likelihood estimates. It's a language problem for how to deal with words like "now" in a consistent way, not a problem with the applicability of probability.

Rather than using conditioning, what we usually do (correctly) is to know that, by itself, the passage of time doesn't change things that don't change with time.

But that just doesn't seem true. If today is Tuesday, she knows that the coin toss result was heads. So knowing it's Tuesday changes the odds from whatever to 1.

Now, if you did really want to condition on the time, you could treat the time as if it was randomly selected. That's fine. Then you would be conditioning on "it is time X and time X was randomly selected". Using this method you would get P(heads) = 1/2 and P(heads|monday) = 2/3.

How do you get that? That's truly nonsensical, for the following reason (pointed out by @PeroK): If the memory wipe happens on the morning of the awakening, right before sleeping beauty wakes up, then there is no need to even toss the coin until Tuesday morning. So on Monday, the coin hasn't even been tossed (under this variant). How could the knowledge that today is Monday tell you about a coin that has not yet been tossed?

It makes a lot more sense to say that knowing it's Monday doesn't tell you anything about the likelihood of a coin toss in the future. So $P(H | Monday) = 1/2$. Knowing that it's Tuesday tells you exactly what the coin toss result was: $P(H | Tuesday) = 1$. If you don't know whether it's a Monday or a Tuesday, then the probability of heads would be: $P(H) = P(H|Monday) P(Monday) + P(H|Tuesday) P(Tuesday) > 1/2$

But thirders are trying to have it both ways. They do not want to treat time as randomly selected or static, yet they want to condition on it.

If you want to randomly select a time, as well as randomly select heads or tails, then there are 4 possibilities, equally likely:

1. Heads and Monday
2. Heads and Tuesday
3. Tails and Monday
4. Tails and Tuesday

Then you rule out 4 on the basis that the rules say no memory wipe in that case. (So if Sleeping Beauty doesn't know whether it's Monday or Tuesday, then she can't be in situation 4). Eliminating situation 4 still leaves the other three equally likely.

 

[stevendaryl]

A reasonable person don't wager.
Look, for once, I'll try to be dead serious. This problem is in the "positive trolling" category, exactly like the chicken and eggs. Those problems have perfectly fine and simple answer, until you add some "hidden" confusion based on tricky language where everyone feels untitled to plug their own meaning.
In sleeping beauty, the trolling start when you speak about days name. This is irrelevant because of the drug and the closed room.

So if the coin is fair, and the experiment is fair, Beauty does not even have to be put to sleep to answer the question. Actually the coins does not even have to be flipped before Tuesday. So asking the question on Monday is definitely some sort of lying if the experimenter choose to do so.I agree, but then it is irrelevant to the question "What is your credence now for the proposition that the coin landed heads?".I have no idea how Beauty or anyone else could do that calculation. "Waking up" have never changed anything to how you process the world, except maybe on the February 2 (arghh I lied, I am not serious anymore)If the question was "how many time do you wake up", I would agree Beauty should answer 3/2.
But that is not the question, isn't it ?

Well, what is your calculation for this modified Sleeping Beauty problem?

1. She gets a memory wipe in either case (heads or tails).
2. But if the coin result was tails, then on Tuesday she is given a note saying that it is, in fact, Tuesday.

Then before she checks whether there is a note, then do you agree that there are 4 possibilities, equally likely?

1. Result was heads, and today is Monday?
2. Result was heads, and today is Tuesday?
3. Result was tails, and today is Monday?
4. Result was tails, and today is Tuesday?

Just from symmetry, wouldn't you give all 4 possibilities equal likelihood?

Now, she checks for a note. If there is no note, then she knows she is in situation 1-3. The lack of a note eliminates situation 4, but it shouldn't affect the likelihood of the other 3 possibilities.

 

[PeterDonis]

for me, relative frequency is equivalent to probability

If you are dealing with a large enough sample space for the law of large numbers to be relevant, then this makes sense. So if Beauty were told that the experiment would be run, say, 1000 times, and after each run ended she would be given the amnesia drug so she would never know, when awakened, which run it was, then using the relative frequency to conclude that P(Monday|Awakened) = 2/3 and P(Tuesday|Awakened) = 1/3 (using my notation from previous posts) would make sense. And that would give P(Heads) = 1/3.

But the original statement of the problem doesn't say that; it says the experiment gets run once and that's it. And I don't think it is a slam dunk that relative frequencies are equal to probabilities if we just have a single sample. The frequentists say it is, but not everyone is a frequentist.

From a Bayesian point of view, if we're only making a single run of the experiment, then the only basis we have for assigning any value at all to P(Monday|Awakened) and P(Tuesday|Awakened) is prior probabilities. If we say that, given only a single run of the experiment, we have no useful information about the distribution of awakenings, then the obvious prior to use is the maximum entropy prior, which is P(Monday|Awakened) = P(Tuesday|Awakened) = 1/2. And if we use those values, then, by my previous posts, we get P(Heads) = 1/4! So this argument is an argument against both the thirder and the halfer. (And we would need to add another option to the poll. )

 

[PeroK]

If you are dealing with a large enough sample space for the law of large numbers to be relevant, then this makes sense. So if Beauty were told that the experiment would be run, say, 1000 times, and after each run ended she would be given the amnesia drug so she would never know, when awakened, which run it was, then using the relative frequency to conclude that P(Monday|Awakened) = 2/3 and P(Tuesday|Awakened) = 1/3 (using my notation from previous posts) would make sense. And that would give P(Heads) = 1/3.

But the original statement of the problem doesn't say that; it says the experiment gets run once and that's it. And I don't think it is a slam dunk that relative frequencies are equal to probabilities if we just have a single sample. The frequentists say it is, but not everyone is a frequentist.

From a Bayesian point of view, if we're only making a single run of the experiment, then the only basis we have for assigning any value at all to P(Monday|Awakened) and P(Tuesday|Awakened) is prior probabilities. If we say that, given only a single run of the experiment, we have no useful information about the distribution of awakenings, then the obvious prior to use is the maximum entropy prior, which is P(Monday|Awakened) = P(Tuesday|Awakened) = 1/2. And if we use those values, then, by my previous posts, we get P(Heads) = 1/4! So this argument is an argument against both the thirder and the halfer. (And we would need to add another option to the poll. )

That analysis doesn't essentially depend on the peculiarities of this experiment. So, that is a critique of probability theory more generally, and specifically in the case where there is only one experiment.

 

[PeterDonis]

that is a critique of probability theory more generally, and specifically in the case where there is only one experiment.

Yes, that's true.

 

[Boing3000]

If you know how to write a computer programme, you could do a computer simulation of the problem and you would, to your surprise, see the answer 1/3 appear rather than 1/2.

I have encoded your logic (if I get it right, your toss 5 & 6 does not follow the same rule)
This is javascript push F12 and cut and paste it into your console

(function BeautyThirders(run) {
    var counts = { H: 0, T: 0 }
    var experiments = [
        function Head(counts) {
            counts.H++;
        },
        function Tail(counts) {
            counts.T++;
            counts.T++;
        }
    ];
    while (--run >= 0) {
        experiments[Math.floor(Math.random() * experiments.length)](counts);
    }
    alert("H/T = " + (counts.H / counts.T).toFixed(2));
})(10000);

So, in that little simulation, we ended up with 3 Heads and 3 Tails for the coin. Beauty was woken 9 times (6 times on a Monday and 3 on a Tuesday). When she was woken, it was as a result of Heads 3 times and Tails 6 times.

Of course, but I am afraid you miss my point about this debate/poll. (I voted for the third option)
Beauty cannot count. She is drugged especially to avoid her to do so.
Given the procedure described into the Wikipedia article the situation is simple and straightforward. The credence that a coin lands Head (or tail for that matter) does not change if Beauty goes to sleep or not.

Therefore, if she counted each time she was woken whether it had been a Head or a Tail, she would have got 6-3 tails. And, that is what's called a relative frequency and is the basis of what a probability is.

She cannot count, so unless some modification is done to the definition of the problem, I reject this arguments.

So, halfers don't allow two bets on the same coin, but thirders do.

That's also how I interpret the difference and why I vote for the latest possibility.
When people whose bread and butter is probability cannot even agree on a simple problem, then my credence that this problem is badly defined is above 99.9%

Finally, the thirders position is: if you don't allow the Tuesday bets to stand, why have them in the first place? Isn't that then a different problem?

Precisely, why indeed have question mark in the definition ? Why speak about bet when there is none ? The credence is 1/2. Now indeed, choosing Head instead of Tail may make you loose twice (an hypothetical bet), and that's why the thirder's view exist.

And that's how the problem should be fixed. The credence that the coin is Head(or Tail) has nothing to do with the credence of loosing one or two bets.
Here is a modified version of the code, which is bet oriented, and don't drug Beauty.
The frequency Beauty bets on Head is the second parameter (1 (always),2(half), 3 ..., > tries == never). So obviously, if you want to win, better never bet heads

(function BeautyBets(run, frequency) {
    var counts = { WakeUps : 0, Wins : 0 }
    var experiments = [
        function Head(counts) {
           //Monday
            if (++counts.WakeUps % frequency == 0)
                counts.Wins++;
            else
                counts.Wins--;
        },
        function Tail(counts) {
           //Monday
            if (++counts.WakeUps % frequency != 0)
                counts.Wins++;
            else
                counts.Wins--;
           //Tuesday
            if (++counts.WakeUps % frequency != 0)
                counts.Wins++;
            else
                counts.Wins--;
        }
    ];
    while (--run >= 0) {
        experiments[Math.floor(Math.random() * experiments.length)](counts);
    }
    alert(counts.Wins.toString() + "wins for " + counts.WakeUps + " bets (" + (counts.Wins / counts.WakeUps).toFixed(2) +")");
})(10000, 3);

 

[PeroK]

Given the procedure described into the Wikipedia article the situation is simple and straightforward. The credence that a coin lands Head (or tail for that matter) does not change if Beauty goes to sleep or not.

Perhaps not, but it does change when the experimenter looks at it. It changes from 1/2 to either 0 or 1.

When Beauty is wakened the coin is not 50-50 heads or tails. It is either definitely Heads or definitely Tails.

Her credence is now definitely wrong, and she knows that. So she must, if she can, look for clues to help her guess what the coin really is.

 

[stevendaryl]

Given the procedure described into the Wikipedia article the situation is simple and straightforward. The credence that a coin lands Head (or tail for that matter) does not change if Beauty goes to sleep or not.

Once again, if instead of wiping her memory in the case of heads, suppose they wipe her memory on Monday regardless, but if the coin flip was tails, her memory is restored 10 minutes after wakening.

Then before the 10 minutes is up, do you agree that Sleeping Beauty has to consider all four of the following possibilities equally likely?

1. It's Monday, and the result was heads.
2. It's Monday, and the result was tails.
3. It's Tuesday, and the result was heads.
4. It's Tuesday, and the result was tails.

After the 10 minutes is up, she knows whether she is in case 4 or not. If case 4 is ruled out, then what are the updated probabilities for 1-3?

 

[Boing3000]

Her credence is now definitely wrong, and she knows that.

Wrong how ? What happens that could make her change her mind about how coins works ?

So she must, if she can, look for clues to help her guess what the coin really is.

No she does not. But if the problem definition is changed, not just to include vaguely some asymmetrical bet, but to attach some new credence to how she must bet to win. This is not equivalent to what the coin is.
Drugged or not, she knows the bet is asymmetrical, she knows that even though coins works 1/2, that she will not be asked about that, but asked about "being awake in the context of this asymmetrical setup" what do you bet ?

So one last time. The thirder are not reading a problem about "what the coin really is". They are reading a problem about what to say about a coin to win some bet
I have trivially modified the code so you can change the amount bet in each case. And non surprisingly if Head bet value is more than twice Tail bet, then it became more "interesting" to always bet on Head. Yet still nothing have happened to the coins, frequencies of days and what not.
It is also not surprising that the only invariant of this algorithm is when frequency is 2 (1/2). Meaning 0 gain or loss. Meaning the only "physical" thing is a fair coin have two side.

(function BeautyBets(run, frequency, headAmount, tailAmount) {
    var counts = { Bets: 0, Wins: 0, Flips: 0 }
    var experiments = [
        function Head(counts) {
            counts.Flips++;

            if (++counts.Bets % frequency == 0)
                counts.Wins += headAmount;
            else
                counts.Wins -= headAmount;
        },
        function Tail(counts) {
            counts.Flips++;

            if (++counts.Bets % frequency != 0)
                counts.Wins += tailAmount;
            else
                counts.Wins -= tailAmount;

            if (++counts.Bets % frequency != 0)
                counts.Wins += tailAmount;
            else
                counts.Wins -= tailAmount;
        }
    ];
    while (--run >= 0) {
        experiments[Math.floor(Math.random() * experiments.length)](counts);
    }
    alert("Money " + counts.Wins);
})(10000, 2, 2.1, 1);

 

[stevendaryl]

Wrong how ? What happens that could make her change her mind about how coins works ?

No she does not. But if the problem definition is changed, not just to include vaguely some asymmetrical bet, but to attach some new credence to how she must bet to win. This is not equivalent to what the coin is.

I don't understand what you mean. Surely, if the walls of Sleeping Beauty's room are painted red if the coin flip is heads and blue otherwise (and she knows this), then she wouldn't stick to her 50% likelihood of heads and tails after seeing the wall color. When someone is asked what the likelihood is that something is true, they typically use whatever information that is available.

The real issue is whether she has any information to change her likelihood estimate. I say she does---the fact that she doesn't know whether it is Monday or Tuesday tells her something---it tells her that either:

1. The result was heads, and today is Monday.
2. The result was heads, and today is Tuesday.
3. The result was tails, and today is Monday.

The fourth possibility---that today is Tuesday and the coin flip was tails---is eliminated by the information that she doesn't know whether it's Monday or Tuesday. (She would know in the fourth case).

 

[PeroK]

Wrong how ? What happens that could make her change her mind about how coins works ?

If I say to you that I am going to toss a coin, you will say the probability of heads is 1/2. That's fine.

Then, I toss the coin and look at it but don't tell you. Then I ask you again. You will still say 1/2.

That's the best you can do, but your information is now out of date. I know what the coin is and, to me, your answer of 1/2 is no longer valid. Or, at least no longer up to date.

Now, I could tell you or show you what the coin is and you would be forced to change your credence from 1/2 to 0 or 1.

Alternatively, I could give you some clues about what the coin is. And, using my clues you might change your credence from 1/2 to something else, based on how big a clue I give you.

This is perhaps easier to see with a die than a coin. I could tell you it's an even number and you would change your credence that it is a 6, say, from 1/6 to 1/3 etc.

The real point of this puzzle is to work out whether there are any clues in the experiment for Beauty to pick up on.

The halfers say there are no clues, so they stick with 1/2.

The thirders, however, are much better at spotting clues, so we find these clues and change our credence to 1/3.

 

[Boing3000]

Once again, if instead of wiping her memory in the case of heads, suppose they wipe her memory on Monday regardless, but if the coin flip was tails, her memory is restored 10 minutes after wakening.

Then before the 10 minutes is up, do you agree that Sleeping Beauty has to consider all four of the following possibilities equally likely?

1. It's Monday, and the result was heads.
2. It's Monday, and the result was tails.
3. It's Tuesday, and the result was heads.
4. It's Tuesday, and the result was tails.

No. Case 3 never enter the possibilities.

After the 10 minutes is up, she knows whether she is in case 4 or not. If case 4 is ruled out, then what are the updated probabilities for 1-3?

Probabilities of coins flipping does not update with knowledge. If she have to bet, she will not change her bet anyway. There is a specific wining strategy in all cases.
I think the halfers are aware that "being awake AND be Tails" is twice that of "being awake AND Heads".
Just change the question so it exclude the confusion with: "being awake AND be the SAME tail as yesterday"

 

[stevendaryl]

No. Case 3 never enter the possibilities.

What do you mean? Are you saying that case 3 never happens, or that it has no associated likelihood?

Probabilities of coins flipping does not update with knowledge.

That is just wrong. When you acquire new information, you update the likelihood of what is true.

If heads versus tails affects what Sleeping Beauty sees, then she can update her estimate of the likelihood based on what she sees. That's the way probabilities work. If someone flips a coin, and they win a prize for heads, and they see the coin but you don't, you can update the likelihood that it is heads by whether they smile or frown. If you know that your uncle just bought a $100,000 sports car, then your estimate of the likelihood that he won the lottery would go up.

The subjective likelihood of an event in the past changes depending on what you know. Are you disagreeing with that? You think that since winning the lottery has a 1 in a million chance, that even if your uncle tells you "I just won the lottery", you won't believe him?

 

[stevendaryl]

Why can't you answer the question:

In the case where the memory restoration happens 10 minutes after Sleeping Beauty wakes up on Tuesday if the result was tails, do you agree that before the 10 minutes is up, that Sleeping Beauty would consider all 4 of the following to be possible?

1. It's Monday, and the result was heads.
2. It's Monday, and the result was tails.
3. It's Tuesday, and the result was heads.
4. It's Tuesday, and the result was tails.

 

[Boing3000]

Now, I could tell you or show you what the coin is and you would be forced to change your credence from 1/2 to 0 or 1.

No. Only me looking at the coins (both side to be sure it is fair) will turn my credence into event/certitude.

Alternatively, I could give you some clues about what the coin is. And, using my clues you might change your credence from 1/2 to something else, based on how big a clue I give you.

I understand that perfectly. And that have nothing to do with coins flipping. It as everything to do with coins "flipping AND more...."

The real point of this puzzle is to work out whether there are any clues in the experiment for Beauty to pick up on.

And yet the very layout of the experiment goes out of his way to drug Beauty so has to be sure "there is no clues" to be gained in the experiment. I make it clear in my first post that there is not even the need to run it. Unless I misunderstood the usage of drug. Was it suppose to make here forget only about the day, or about the whole setup explanation ?

The halfers say there are no clues, so they stick with 1/2.

Indeed, the halfers says that, as well as the very definition of the problem. And they are dam right about 1/2 of Head actual counts.

The thirders, however, are much better at spotting clues, so we find these clues and change our credence to 1/3.

There are not clues to spot. All data are present before hand, and there is 2/3 chances to be awake on tail.

I say: both group don't answer the same question.

 

[stevendaryl]

Probabilities of coins flipping does not update with knowledge. If she have to bet, she will not change her bet anyway.

That doesn't make any sense. In the revised experiment (on Tuesday, if the result was tails, her memory is restored 10 minutes after she wakes), she wakes up knowing that she's in one of 4 situations:

1. Heads and Monday
2. Heads and Tuesday
3. Tails and Monday
4. Tails and Tuesday

If you asked her to bet on which situation is true, what odds would she take? I can't see any reason for her not to give all four possibilities equal likelihood.

Now, in case #4, she finds out that she is in case #4. In cases 1-3, she finds out only that she is not in case #4. Now, what likelihood would she give to each of the possibilites 1-3? How much would she be willing to bet on each?

 

[stevendaryl]

And yet the very layout of the experiment goes out of his way to drug Beauty so has to be sure "there is no clues" to be gained in the experiment.

But that's not true. When she wakes up, she finds out that she is not in situation #4 (it's Tuesday, and the result was tails). (edit: I originally said "heads")

 

[Boing3000]

Why can't you answer the question:

In the case where the memory restoration happens 10 minutes after Sleeping Beauty wakes up on Tuesday if the result was tails, do you agree that before the 10 minutes is up, that Sleeping Beauty would consider all 4 of the following to be possible?

1. It's Monday, and the result was heads.
2. It's Monday, and the result was tails.
3. It's Tuesday, and the result was heads.
4. It's Tuesday, and the result was tails.

Beauty has been explained all the possibilities and 3 is not one of them.
From wikipedia: " if the coin comes up heads, Beauty will be awakened and interviewed on Monday only"

That is just wrong. When you acquire new information, you update the likelihood of what is true.

I am told that a fair coins is 1/2 chance to be head. And frankly, I tend to believe it, because it matches my experiences so far.
Then I am told that same coins have just run 1 millions head in a row. Now, I know it is unlikely, but that newly acquired information change nothing. If that coin is fair then it is still 1/2.

 

[Boing3000]

But that's not true. When she wakes up, she finds out that she is not in situation #4 (it's Tuesday, and the result was tails). (edit: I originally said "heads")

Maybe we haven't read the same setup. Mine if from wikipedia, and she cannot finds anything by construction of the experiment:
wiki: "Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Beauty is asked: "What is your credence now for the proposition that the coin landed heads?".

 

[stevendaryl]

Beauty has been explained all the possibilities and 3 is not one of them.
From wikipedia: " if the coin comes up heads, Beauty will be awakened and interviewed on Monday only"

Hmm, I think I may have gotten heads and tails confused, in that case. The way I have been assuming is that

• If it's heads, she is woken up on Monday and Tuesday
• If it's tails, she is woken up on Monday
• And the twist: If it's tails, of course she will wake up on Tuesday, but she will wake up knowing that it is in fact Tuesday.

I am told that a fair coins is 1/2 chance to be head. And frankly, I tend to believe it, because it matches my experiences so far.

Yes, but if I further tell you that when it lands heads, I dance a jig, and I don't dance a jig, then your likelihood changes. Likelihood of the truth of something changes based on new information. That's what probability is good for---taking into account new information.

 

[stevendaryl]

Maybe we haven't read the same setup. Mine if from wikipedia, and she cannot finds anything by construction of the experiment:

But if she doesn't know what day it is, then that is itself information, because she would know what day it was if it were Tuesday and the coin toss had been a particular result (whichever result causes her memory not to be wiped).

 

[PeroK]

No. Only me looking at the coins (both side to be sure it is fair) will turn my credence into event/certitude.

.

You are misunderstanding the point of the problem. We are not discussing here the situation with conjurers, liars or unfair coins. That is a completely different problem altogether.

In that case, none of the answers or analyses given here are valid.

If a stage magician tossed a fair coin there is a finite probability that it would come up as a playing card showing the ace of spades.

 

[Boing3000]

Hmm, I think I may have gotten heads and tails confused, in that case. The way I have been assuming is that

• If it's heads, she is woken up on Monday and Tuesday
• If it's tails, she is woken up on Monday
• And the twist: If it's tails, of course she will wake up on Tuesday, but she will wake up knowing that it is in fact Tuesday.

OK, but I suppose we'd better stick to the actual Beauty "problem" which is not what you describe. And btw the "twist/drug" is precisely that beauty will NOT be able to tell what day it is (or if it is the millionth time she is woken (well if there is no mirror in the room at least ))

Yes, but if I further tell you that when it lands heads, I dance a jig, and I don't dance a jig, then your likelihood changes.

Not at all. Only the likely hood that you dance a jig increase (even though I am quite sure you are a happy guy on average ). And the information don't concerns coins tossing, but you dancing. You make "a promise" to me (that I would assign some probability you would respect)

Likelihood of the truth of something changes based on new information.

That's a agreeable statement but it is unlikely to be true. The precise exact (mathematical) correlation between information is what truth means for me.

That's what probability is good for---taking into account new information.

In my book, probability is the tool to manage absence of information. Everybody knows there is no randomness in coins tossing (the lower bound of that ignorance is chaos) and only our absence of knowledge of the "exact" way the coin is tossed, force us to use probability tool to manage the truths about the situation.

New information (like dancing, or wakening) added linearly to the consequences of that original "fuzziness" don't change that original "fuzziness"