The Sleeping Beauty Problem: Any halfers here?

[Marana]

@Marana If you were the sleeper, please tell me what you would answer to these three questions:

You wake up:

What is the probability that the coin is heads?

If it's Monday, what would be your answer?

If it's Tuesday, what would be your answer?

Probability that the coin is heads: 1/2

If it's Monday: 1/2

If it's Tuesday: 0

Or, someone has two children. The probability of two boys is 1/4.

If they have two girls then they come to see you on a Monday; otherwise, they come to see you on a Tuesday. Nothing random. Yet, if they come to see you on a Tuesday, the probability of two boys has increased to 1/3.

Probability is not only about randomness, but also about absence of knowledge. Suppose that I pick one of the letters A or B, by will. Then I ask you, what is the probability that I picked A? What is your answer?

Here's the way that I became a thirder, which I think is convincing (even if it is much more work than the original, one-line argument for 2/3 or 1/2).

Imagine that experimenters are doing this experimenter over and over, with lots of different test subjects (sleeping beauties).

It's not just the randomness that concerns me, it is whether it is an experiment. "In probability theory, an experiment or trial is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space."

I can select a new two-child family each week. The experimenters can select a random beauty each day. Both are easily repeatable and have a clear probability. As for picking a letter, I don't believe I could put an exact number on the probability (principle of indifference would say 1/2, but I'm not totally indifferent as I'd guess A is more popular, similar to how certain numbers show up more than others).

The difficulty with the sleeping beauty problem is that waking up isn't an experiment at all. It is, by definition, impossible to repeat. Tuesday follows Monday by the laws of the universe, TT follows MT by the laws of the study. A single waking is insufficient to model the situation, the rules of which are known to sleeping beauty.

"It is Monday" and "it is Tuesday" can both be learned for a single coin flip. That is not consistent with conditioning. So if we are asked about the result of the coin flip, it isn't justified to condition on the day of the week. It only seems reasonable at first because of the memory loss.

So I'd begin with the coin flip, a random experiment with sample space {H, T} and probability 1/2 for each. When I wake up I would maintain probability 1/2 for various reasons (lack of new relevant info, principle of reflection, intuition due to thirders being able to all believe they won billion dollar lottery) while admitting none of those reasons are fully convincing, just more convincing than the alternative. Then if I learn "it is Monday" I will recall that I may also learn "it is Tuesday", so that this is not the kind of thing I can use conditioning on. Time marches on, and if it is Monday, that means the probability is 1/2. Either because the coin doesn't need to be flipped yet, or because Monday tails is the precursor to Tuesday tails (really "MT followed by TT" as a whole is an outcome of the coin flip experiment).

 

[PeroK]

Tuesday follows Monday by the laws of the universe.

Not if you've been given an amnesia drug. Then "I don't know what day it is" is followed by "I don't know what day it is". Only those with a good memory, or who can look up the information, know what day it is.

 

[Demystifier]

but I'm not totally indifferent as I'd guess A is more popular

And I know that, so to deceive you I will be more prone to choose B. But I know that you know that too, so I will deceive you at a higher level by being more prone to choose A. But I know that you know that I know that you know that, so perhaps I should deceive you at an even higher level be being more prone to choose B ... When one takes all this into account, A and B seem about equally likely.

 

[stevendaryl]

It's not just the randomness that concerns me, it is whether it is an experiment. "In probability theory, an experiment or trial is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space."

You quoted one line of my post, but I don't see that you responded to it. Do you agree that in the setup I described, it makes sense for an observer to assign a 2/3 probability that a randomly selected sleeping beauty has an associated coin flip result of heads? The way I described that thought-experiment seems perfectly amenable to usual probabilistic reasoning. Right?

The next step would be for each sleeping beauty herself to consider herself a random choice. She knows that there are $3N/2$ people in the same situation she is in--not knowing whether they have been awakened one time, or two. Of those, she knows that

• $N/2$ had a coin toss result of heads, and are awakening for the first time.
• $N/2$ had a coin toss result of heads, and are awakening for the second time.
• $N/2$ had a coin toss result of tails, and are awakening for the first time.

So it seems completely straight-forward that she would assume that

• Her probability of having a result of heads is 2/3
• The probability that she is being awoken the first time is 2/3
• The probability that she is being awoken the second time is 1/3.

The final logical leap is to assume that the probabilities that apply in a repeated experiment also apply in a single experiment.

 

[stevendaryl]

And I know that, so to deceive you I will be more prone to choose B. But I know that you know that too, so I will deceive you at a higher level by being more prone to choose A. But I know that you know that I know that you know that, so perhaps I should deceive you at an even higher level be being more prone to choose B ... When one takes all this into account, A and B seem about equally likely.

That's a standard result in game theory, called a "mixed strategy". In a game like chess, there is always a "best move" (or perhaps a number of equally good moves), and there is no reason not to make it. But in certain types of games, your best move is to be random. An example is "Battleship".

 

[stevendaryl]

The probability for Monday is 3/4 and for Tuesday 1/4.

Let me try another variant of the experiment that I think will convince you that you're wrong.

Have you seen the movie "Memento"? The main character has a form of amnesia where he wakes up every morning having no idea what happened the previous day, unless he left notes for himself beside his bed (or pinned to his pajamas, or whatever). So we can redo the Sleeping Beauty problem using such an amnesiac. There is no need to wipe memories, but instead, we just control what notes she has waiting for her on the two mornings, Monday and Tuesday.

We prepare two envelopes. The first envelope says "Read me first" on the outside. Inside is a note explaining the rules of the experiment. Regardless of the coin toss, she gets this note on both Monday and Tuesday. The second envelope says "Read me second" on the outside. Inside is a note saying either "Today is Tuesday" or "Today is either Monday or Tuesday". She only gets the note saying "Today is Tuesday" if it actually is Tuesday, and the coin toss result was tails.

She wakes up and reads her first envelope, and is asked her subjective probabilities for various things. I believe everyone would agree that her answers would be:

1. There is a 1/4 chance that it's Monday and the coin toss result was Heads.
2. There is a 1/4 chance that it's Monday and the coin toss result was Tails.
3. There is a 1/4 chance that it's Tuesday and the coin toss result was Heads.
4. There is a 1/4 chance that it's Tuesday and the coin toss result was Tails

All four situations are equally likely.

Now, she reads the second note and finds out that she is not in situation 4. The other three situations are still equally likely, though.

 

[Charles Link]

@stevendaryl Thank you. :) Very interesting, but this is one of those that I think I could study for years and not be convinced that there is one answer that is completely correct. It's like the riddle of "Who's on first. (baseball=first base). Is "Who" on first, etc.? It comes up in the Dustin Hoffman, Tom Cruise movie "Rain Man", and Tom Cruise tells his brother that it is a riddle that has no correct answer. :) :)

 

[PeroK]

@stevendaryl Thank you. Very interesting, but this is one of those that I think I could study for years and not be convinced that there is one answer that is completely correct. It's like the riddle of "Who's on first. (baseball=first base). Is "Who" on first, etc.? It comes up in the Dustin Hoffman, Tom Cruise movie "Rain Man", and Tom Cruise tells his brother that it is a riddle that has no correct answer. :) :)

Hmm. I'm not prepared to give up on mathematics just because a problem is not intuitively obvious. If we do that, then we are left with nothing. If there is a compelling argument that probability theory cannot be used in this case, then I'm happy to listen and accept that the problem has no solution. I've not seen any such argument yet, I have to say.

 

[PeterDonis]

It looks like I'm the only vote in the poll for "it depends on the precise formulation of the problem", which seems strange, since the fact of this thread going on for 6 pages would seem to be evidence in favor of that choice. (A better basis for it, though, IMO is Demystifier's post #67.)

 

[PeroK]

It looks like I'm the only vote in the poll for "it depends on the precise formulation of the problem", which seems strange, since the fact of this thread going on for 6 pages would seem to be evidence in favor of that choice. (A better basis for it, though, IMO is Demystifier's post #67.)

I looked again at post #67. I cannot see any reason that in this experiment we would only count one of the instances in the case of tails. Yes, we can change the problem to say that we will deselect one of the tails instances. But that reduces the problem to something really quite trivial. I cannot see that in the case of rule B there is anything worth analysing or discussing.

In other words, the answer of 1/2 only applies in a case where the problem is so trivial as to merit no attention.

But, what makes the problem interesting is to explain the apparent paradox that the answer of 1/3 arises after "no new information". It is a seductive argument but we have clearly identified a change to the sleeper's information:

She no longer knows what day it is. And, in fact her information precisely coincides with a "random observer" who randomly stumbles over the experiment and doesn't know what stage the experiment has reached.

Finally, we could easily have six pages of argument about whether $0.999 \dots \ne 1$ - or whether simultaneity is absolute. But, it doesn't matter how long and hard someone argues either of those points or that the answer to the sleeping beauty problem is 1/2. Pertinancity alone doesn't make their case.

 

[PeterDonis]

I cannot see any reason that in this experiment we would only count one of the instances in the case of tails.

Um, because the experimenter decided to define the experiment that way? The scenario described in post #67 is about betting, and a bet can be whatever the bettors want it to be. Unless you're claiming that it's somehow physically impossible for an experimenter to offer Beauty the bet described by rule B in post #67.

Basically, you're trying to go from "I can't see any reason..." to "obviously my answer is the only right answer". But that's not a valid argument, logically speaking; it's just a statement of your opinion. There is no unique right answer until you've specified a question that's precise enough to have a unique right answer. Post #67 simply illustrates one of the things that has to be part of that precise specification in order for your answer to be the unique right answer.

 

[PeroK]

Um, because the experimenter decided to define the experiment that way? The scenario described in post #67 is about betting, and a bet can be whatever the bettors want it to be.

Then that is a very different problem from the one (in my opinion) precisely described in the original problem statement. The original problem statement (from the experimenters' point of view) is clear. If it's heads, they wake her on the Monday and if it's tails on the Monday and the Tuesday.

It says nothing in the original problem statement about possibly waking her on the Tuesday if it's tails or possibly not waking her on the Monday if it's tails.

I don't really see how that could be any clearer. Post #67, I believe, only introduced this variation to highlight that the interpretation inferred by the 1/2 argument was a different problem altogether.

In fact, if you start with the trivial problem that they wake her only on the Monday and hence the answer is 1/2, then the halfer position is, effectively, that whatever happens on subsequent days is irrelevant.

You can see this more clearly in the $n, k$ variation, where she is woken on $n$ consecutive days (heads) and $k$ consecutive days (tails). The 1/2 answer remains 1/2 in all cases. The 1/3 answer is $n/(n+k)$.

In other words, the halfer position in general only considers the first day. Or changes the problem so that only one of the $n$ days and one of the $k$ days "counts". That is a different problem altogether.

Furthermore, the only thing that makes this problem difficult is the amnesia drug. The 1/2 position can only arise in the case of an amnesia drug. Without the drug, the problem is trivial and the answer is 1/3. And, I suggest, that without the drug, no one would be claiming that the problem is not precise.

Finally, I see the fundamental difference in this thread as the 1/3 position has been backed up by analysis; whereas, it is the 1/2 position that is largely opinion: we can't use relative frequencies; we can't use a Bayesian approach; it's not a random experiment; there is no sample space; conditional probabilities don't apply; etc.

In each case, an analysis of why these objections are invalid has been presented, although I guess now the core analysis has been lost in 6 pages of claim and counterclaim.

 

[PeroK]

I think it's interesting to quote Wikipedia on this:

"The Sleeping Beauty puzzle reduces to an easy and uncontroversial probability theory problem as soon as we agree on an objective procedure how to assess whether Beauty's subjective credence is correct. Such an operationalization can be done in different ways: By offering Beauty a bet; more elaborately by setting up a Dutch book; or by repeating the experiment many times and collecting statistics. For any such protocol, the outcome depends on how Beauty's Monday responses and her Tuesday responses are combined.

Consider long-run average outcomes. Suppose the experiment were repeated 1,000 times. It is expected that there would be about 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday.

• If Beauty herself collects statistics about the coin tosses (in a way that is not obstructed by memory erasure when she is put back to sleep), she would register one-third of heads. If this long-run average should equal her credence, then she should answer P(Heads) = 1/3.
• However, being fully aware about the experimental protocol and its implications, Beauty may reason that she is not requested to estimate statistics of the circumstances of her awakenings, but statistics of coin tosses that precede all awakenings. She would therefore answer P(Heads) = 1/2.

It's even simpler with bets: If Beauty and the experimenter agree that bets from her different awakenings are cumulative, then a heads quota of 1/3 would be fair. If on the other hand Tuesday bets are to be discarded (being dummy bets, undertaken only to keep Monday and Tuesday awakenings indistinguishable for Beauty), then the fair quota would be 1/2."

That is, then, fairly explicit on the circumstances under which 1/2 is a valid answer. I've highlighted one sentence. My question is this: why would one calculate a probability on that basis? Without the drug, the second operationalisation is not logical at all. It's Tuesday, you are asked to estimate the probability that the coin was heads. You know it's tails, but, nevertheless, you answer 1/2 because you are still thinking about the coin before any awakenings?

With the drug, you have some information (you are not sure what day it is), but for some reason you don't or can't use that information. You simply revert to the pre-experiment answer?

So, it seems the second operationalisation (halfer position) works on the basis of: if your knowldege is imperfect then yoiu ignore that knowldege and the answer is 1/2. Only if you definitely know the result can you answer 1 or 0. But, nothing in between.

(But, this only applies in the case of drug-induced memory loss. Outside of this special case, halfers revert (hopefully) to normal probability theory where you use all the information at your disposal. So, for halfers this is very much a one-off special case where the normal rules of probability theory do not apply.)

This isn't probability theory as I understand it. I believe you are free to use all the information you have to calculate a probability. And, if you deliberately or unwittingly don't use information at your disposal, then you are not calculating the probability as I would understand it.

And, by using all the information you have about the experiment, that you have been woken, that you don't know what day it is (it could be Tuesday and it could definitely be tails), then you get an answer of 1/3.

If you do not use this information, then you can get an answer of 1/2, but that is not the probability of its being heads, given everything you know.

PS And, if the answer is 1/2, then you can deduce with certainty that it is Monday. Proof:

Suppose the probability that it is heads is 1/2 and the probability it is Monday is $p$. Then the probability it is heads is:

$\frac{p}{2} + 0 = \frac{p}{2} = 1/2$

Hence, $p = 1$.

Therefore, whatever the answer is, it cannot be 1/2, unless the problem is changed so that Tuesday is excluded.

 

[Marana]

Furthermore, the only thing that makes this problem difficult is the amnesia drug. The 1/2 position can only arise in the case of an amnesia drug. Without the drug, the problem is trivial and the answer is 1/3. And, I suggest, that without the drug, no one would be claiming that the problem is not precise.

Finally, I see the fundamental difference in this thread as the 1/3 position has been backed up by analysis; whereas, it is the 1/2 position that is largely opinion: we can't use relative frequencies; we can't use a Bayesian approach; it's not a random experiment; there is no sample space; conditional probabilities don't apply; etc.

In each case, an analysis of why these objections are invalid has been presented, although I guess now the core analysis has been lost in 6 pages of claim and counterclaim.

Nobody would be answering 1/3 without the drug. Without the drug the answer is 1/2 on monday and 0 on tuesday (because you keep your memory you always know what day it is).

Indeed, that illustrates why the frequency argument fails. Without the drug, the frequencies are the same, but not a single person would answer 1/3. The fact that it is not a repeatable random experiment, and not solvable by frequency, would be crystal clear.

That leaves us with the conditioning method, but I can demonstrate that it clearly fails as well. All I have to do is mention that you can learn both "it is monday" and "it is tuesday" for a single coin flip, totally impossible for conditioning. Therefore you can't condition on the day when trying to decide on the result of the coin flip.

We need a different way to compute probabilities with information like "it is monday" and "it is tuesday" that can't be used with conditioning. My idea so far is that if you have a correspondence of non-temporal information between "starting time" and "some specified later time", and you learn only that "some specified later time" is true, then your probabilities in things that don't change with time (like the result of a coin flip) must be equal to what they were at the starting time. So in sleeping beauty, there is a correspondence between yourself on sunday and yourself on monday (because you are always awake monday). If you learn that it is monday your probability should be 1/2, because that is what it was on sunday (the starting time) and the only thing that changed is time (as it inevitably does) marched on.

For a slightly more complicated example, if you wake up every day for a year on tails, and only prime days for heads, and you learn it is day 137, then your probability should be 1/2, because the only change from the starting time is the passage of time, which has no relevance to the result of the coin flip. This method can probably be improved to deal with more complicated situations, the main thing is that conditioning does not apply.

PS And, if the answer is 1/2, then you can deduce with certainty that it is Monday. Proof:

Suppose the probability that it is heads is 1/2 and the probability it is Monday is $p$. Then the probability it is heads is:

$\frac{p}{2} + 0 = \frac{p}{2} = 1/2$

Hence, $p = 1$.

Therefore, whatever the answer is, it cannot be 1/2, unless the problem is changed so that Tuesday is excluded.

I don't think I am being all that radical by saying things like this are not allowed, for the reason I mentioned above. "It is Monday" and "it is Tuesday" are a different kind of information which can't use the usual probability techniques. But I'm not just making that up: it is fact that you may learn both "it is Monday" and "it is Tuesday" for a single flip, and it is a fact that that is absolutely impossible with normal probability techniques.

It's like saying you "learned" it was 2pm. Then you "learned" it was 2:01. Then you "learned" it was 2:02. Well, no, you're not learning all that in the usual sense. Time is just going forward.

 

[PeroK]

Nobody would be answering 1/3 without the drug. Without the drug the answer is 1/2 on monday and 0 on tuesday (because you keep your memory you always know what day it is).

The answer is 1/3 as follows:

$P(H) = P(H|Mon)P(Mon) + P(H|Tue)P(Tue) = (1/2)(1/3) + 0(2/3) = 1/3$

That's the conditional probability of its being Heads given a random awakening.

The specific answers given are 1/2 and 0, but this equates to a conditional probability of 1/3.

The point is that the overall conditional probability in this case is not 1/2.

That's the way conditional probabilities work. In the same way that an average can be a number that it not iself attainable in any experiment, it is also the case with a conditional probability.

I don't think I am being all that radical by saying things like this are not allowed, for the reason I mentioned above. "It is Monday" and "it is Tuesday" are a different kind of information which can't use the usual probability techniques.

You are being totally radical. There is no reason not to use probability theory in this case.

Yours is an extreme position: adopted in support of your a priori requirement that the answer to the sleeping beauty problem is 1/2.

 

[PeroK]

It's like saying you "learned" it was 2pm. Then you "learned" it was 2:01. Then you "learned" it was 2:02. Well, no, you're not learning all that in the usual sense. Time is just going forward.

I fail to see the relevance of this to the problem. If I look at the clock and learn it is 2pm and then I fall asleep and wake up and look at the clock, then I learn that it is 3pm and that I have been asleep for an hour. It's not somehow inevitably 3pm when I wake up any more than it is inevitably 4pm when I wake.

 

[Marana]

You quoted one line of my post, but I don't see that you responded to it. Do you agree that in the setup I described, it makes sense for an observer to assign a 2/3 probability that a randomly selected sleeping beauty has an associated coin flip result of heads? The way I described that thought-experiment seems perfectly amenable to usual probabilistic reasoning. Right?

The next step would be for each sleeping beauty herself to consider herself a random choice. She knows that there are $3N/2$ people in the same situation she is in--not knowing whether they have been awakened one time, or two. Of those, she knows that

• $N/2$ had a coin toss result of heads, and are awakening for the first time.
• $N/2$ had a coin toss result of heads, and are awakening for the second time.
• $N/2$ had a coin toss result of tails, and are awakening for the first time.

So it seems completely straight-forward that she would assume that

• Her probability of having a result of heads is 2/3
• The probability that she is being awoken the first time is 2/3
• The probability that she is being awoken the second time is 1/3.

The final logical leap is to assume that the probabilities that apply in a repeated experiment also apply in a single experiment.

I may not be understanding this argument fully, but it seems like it is very similar to the frequency argument, just with the experiments placed side by side instead of one after the other.

If so, I agree with you from the outside observer perspective, but not from sleeping beauty's perspective.

The outside observer can think of her as a random choice and assign 2/3 probability, but as I see it sleeping beauty can't consider herself a random choice. That's because her waking up is not a random experiment (while the observer picking someone is a random experiment) and therefore hard to justify using any kind of frequency.

Let me try another variant of the experiment that I think will convince you that you're wrong.

Have you seen the movie "Memento"? The main character has a form of amnesia where he wakes up every morning having no idea what happened the previous day, unless he left notes for himself beside his bed (or pinned to his pajamas, or whatever). So we can redo the Sleeping Beauty problem using such an amnesiac. There is no need to wipe memories, but instead, we just control what notes she has waiting for her on the two mornings, Monday and Tuesday.

We prepare two envelopes. The first envelope says "Read me first" on the outside. Inside is a note explaining the rules of the experiment. Regardless of the coin toss, she gets this note on both Monday and Tuesday. The second envelope says "Read me second" on the outside. Inside is a note saying either "Today is Tuesday" or "Today is either Monday or Tuesday". She only gets the note saying "Today is Tuesday" if it actually is Tuesday, and the coin toss result was tails.

She wakes up and reads her first envelope, and is asked her subjective probabilities for various things. I believe everyone would agree that her answers would be:

1. There is a 1/4 chance that it's Monday and the coin toss result was Heads.
2. There is a 1/4 chance that it's Monday and the coin toss result was Tails.
3. There is a 1/4 chance that it's Tuesday and the coin toss result was Heads.
4. There is a 1/4 chance that it's Tuesday and the coin toss result was Tails

All four situations are equally likely.

Now, she reads the second note and finds out that she is not in situation 4. The other three situations are still equally likely, though.

I like this idea, but I don't think it is equivalent any more to the original.

One of the weaknesses with the thirder arguments I've seen is that they don't seem to model all of sleeping beauty's information. Sleeping beauty is well aware of the rules, of the way monday tails and tuesday tails are inextricably linked by the passage of time, and of the week she will next wake up in.

That last part is important in your example. If sleeping beauty never has any memory, then she can't be aware of what week she is going to wake up in. This is added uncertainty. She is lost not only within the week, but between weeks. So it could be argued that she is now more likely to be in a tails week (since they are longer).

 

[PeroK]

I wanted to highlight one more thing from the Wikipedia entry and then I'm done with this:

"Halfer position
David Lewis responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2.[6] Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is P(Heads) = 1/2, she ought to continue to have a credence of P(Heads) = 1/2 since she gains no new relevant evidence when she wakes up during the experiment. This directly contradicts one of the thirder's premises, since it means P(Tails | Monday) = 1/3 and P(Heads | Monday) = 2/3."

But, what if the experimenters don't even look at the coin until the Tuesday? (You could even, I believe, wait until the Tuesday morning before tossing the coin as the result is not needed until then).

Now, according to the halfers, we have the situation where the sleeper can conclude that if it's Monday, then a coin that has not yet been looked at (or not yet even tossed) must be more likely to be heads than tails. And that is absurd.

This, again, suggests to me that the halfers are not dealing in probabilities (or any quantity - call it "credence" - that behaves in a mathematically consistent way). In particular, the answer of 1/2 is not consistent with other calculations relating to the problem and is, therefore, a mathematically meaningless answer.

A crude analogy is to ask someone what the odds are of their team winning a football match. They are, as ever, 100% certain. That is their credence. But, it is not a quantity that stands up to mathematical scrutiny.

 

[stevendaryl]

If so, I agree with you from the outside observer perspective, but not from sleeping beauty's perspective.

The outside observer can think of her as a random choice and assign 2/3 probability, but as I see it sleeping beauty can't consider herself a random choice.

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.

I like this idea, but I don't think it is equivalent any more to the original.

I don't see how it can make any difference between the two possibilities:

1. In the case of tails, the sleeping beauty is never memory-wiped a second time.
2. In the case of tails, the sleeping beauty is memory-wiped, but the next morning, a note restores her memory (by telling her everything that she forgot).

One of the weaknesses with the thirder arguments I've seen is that they don't seem to model all of sleeping beauty's information. Sleeping beauty is well aware of the rules, of the way monday tails and tuesday tails are inextricably linked by the passage of time, and of the week she will next wake up in.

The thirder argument doesn't ignore that. She knows that Tuesday follows Monday. But she doesn't know how much time has passed since the start of the experiment, so she doesn't know whether it is Monday or Tuesday.

That last part is important in your example. If sleeping beauty never has any memory, then she can't be aware of what week she is going to wake up in. This is added uncertainty. She is lost not only within the week, but between weeks.

First of all, I'm assuming that the notes get her caught up by telling her what the situation is, what the rules are, and that it is either a Monday or Tuesday after the beginning of the experiment. You can give more information and say that the experiment began on June 1, 2017. You can give her the entire story of her life. But you leave out the information about whether it is Monday or Tuesday. So I don't understand that objection.

So it could be argued that she is now more likely to be in a tails week (since they are longer).

I don't understand that. Weeks are seven days long, regardless of whether a head or tail was thrown. The only difference is that on Tuesday morning,

• If the coin was tails, her note tells her that it is Tuesday.
• If the coin was tails, her note only tells her that it is either Monday or Tuesday.

She knows the above two rules (because the notes describe the rules), so she can use the fact that her note did not say that it was Tuesday to draw some conclusions: namely, that either today is Monday, or the coin toss was heads.

 

[stevendaryl]

I think that if we transform the problem into one where Sleeping Beauty's answer has consequences (by having her bet on the coin flip, for example), then both halfers and thirders will agree on the answer, provided that the consequences are spelled out in sufficient concrete detail. So to me, that means that there isn't actually a disagreement about mathematics. It's a disagreement about the meaning of probability, when it is abstracted away from consequences.

 

[PeroK]

I think that if we transform the problem into one where Sleeping Beauty's answer has consequences (by having her bet on the coin flip, for example), then both halfers and thirders will agree on the answer, provided that the consequences are spelled out in sufficient concrete detail. So to me, that means that there isn't actually a disagreement about mathematics. It's a disagreement about the meaning of probability, when it is abstracted away from consequences ...

... and doesn't have to be self-consistent or obey the rules that "real" probabilities do.

 

[PeterDonis]

why would one calculate a probability on that basis?

Um, because that's how one interpreted the words "what is your subjective credence that the coin came up heads?"

The issue isn't that the math is unclear once we've decided which math we're using. The issue is that the problem is not stated in math, it's stated in ordinary language, and ordinary language is vague. Once you remove the vagueness by specifying exactly what mathematical calculation corresponds to "subjective credence", of course there's a unique right answer. But you can't just declare by fiat that your preferred mathematical calculation is the only possible one corresponding to the vague ordinary language used in the problem statement. You don't get to decide how other people interpret vague ordinary language.

 

[PeroK]

Um, because that's how one interpreted the words "what is your subjective credence that the coin came up heads?"

The issue isn't that the math is unclear once we've decided which math we're using. The issue is that the problem is not stated in math, it's stated in ordinary language, and ordinary language is vague. Once you remove the vagueness by specifying exactly what mathematical calculation corresponds to "subjective credence", of course there's a unique right answer. But you can't just declare by fiat that your preferred mathematical calculation is the only possible one corresponding to the vague ordinary language used in the problem statement. You don't get to decide how other people interpret vague ordinary language.

That's more that a trifle harsh given the time and effort I've put into analysing this problem. My conclusions are backed up considerably by the Wikipedia and other analyses, which do not dwell on the possible alternative problems, but on the one as stated.

The problem with the alternative calculations is that they are not consistent with what is meant by a probabilty. For example, if someone claims that it's 100% certain that their football team will win, then I can't argue against that, per se. But, it isn't a probability that can be used in any mathematical calculations.

It's the halfers who are arguing by fiat, not me!

The fiat is that the answer to this problem is 1/2 and everything else must fall into line.

 

[PeterDonis]

and doesn't have to be self-consistent or obey the rules that "real" probabilities do.

I don't see this at all. The two answers, 1/3 and 1/2, are both well-defined conditional probabilities, just different ones:

1/3 is the conditional probability that the coin came up heads, given that Beauty has just been awakened and that we are randomly choosing between the three possible conditions under which Beauty can be awakened (Monday heads, Monday tails, Tuesday tails).

1/2 is the conditional probability that the coin came up heads, given that it was flipped.

Your problem is that you simply can't see how the vague ordinary language in the problem statement could lead anyone to think that the second conditional probability was what was meant by "subjective credence that the coin came up heads". But, as I said in my last post, you don't get to decide how other people interpret vague ordinary language. Evidently some people do think it's possible that Beauty could interpret that vague ordinary language as referring to the second conditional probability rather than the first. Others, such as me, think that since the ordinary language is vague, it doesn't refer unambiguously to either conditional probability, and more specification is needed to pin things down.

 

[PeterDonis]

The problem with the alternative calculations is that they are not consistent with what is meant by a probabilty.

See my previous post.

 

[PeroK]

I don't see this at all. The two answers, 1/3 and 1/2, are both well-defined conditional probabilities, just different ones:

1/3 is the conditional probability that the coin came up heads, given that Beauty has just been awakened and that we are randomly choosing between the three possible conditions under which Beauty can be awakened (Monday heads, Monday tails, Tuesday tails).

1/2 is the conditional probability that the coin came up heads, given that it was flipped.

This is not the issue here. That issue would not cause any arguments because those are two very different problems. You've clearly read too little of the background to this problem. What you have stated is not the sleeping beauty dichotomy.

(We certainly haven't spent 7 pages arguing about that!)

Read the Wikipedia page to see the real issue:

https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

 

[PeterDonis]

Read the Wikipedia page to see the real issue

Here is how the Wikipedia page sums up the thirder argument:

"Since these three outcomes are exhaustive and exclusive for one trial, the probability of each is one-third"

In other words, Beauty's "subjective credence" should correspond to the first conditional probability I gave.

Here is how the Wikipedia page sums up the halfer argument:

"Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is P(Heads) = 1/2, she ought to continue to have a credence of P(Heads) = 1/2 since she gains no new relevant evidence when she wakes up during the experiment."

In other words, her "subjective credence" should correspond to the second conditional probability I gave.

So I don't see how I've failed to describe the issue correctly. I've obviously left out a lot of details in each argument, but so what? I've correctly identified the two conditional probabilities, and that was all I was trying to do. The rest of the details are just attempts to justify why one particular conditional probability is the "right" one, the one that the words "subjective credence" should be interpreted to refer to.

 

[PeroK]

Here is how the Wikipedia page sums up the thirder argument:

"Since these three outcomes are exhaustive and exclusive for one trial, the probability of each is one-third"

In other words, Beauty's "subjective credence" should correspond to the first conditional probability I gave.

Here is how the Wikipedia page sums up the halfer argument:

"Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is P(Heads) = 1/2, she ought to continue to have a credence of P(Heads) = 1/2 since she gains no new relevant evidence when she wakes up during the experiment."

In other words, her "subjective credence" should correspond to the second conditional probability I gave.

So I don't see how I've failed to describe the issue correctly. I've obviously left out a lot of details in each argument, but so what? I've correctly identified the two conditional probabilities, and that was all I was trying to do. The rest of the details are just attempts to justify why one particular conditional probability is the "right" one, the one that the words "subjective credence" should be interpreted to refer to.

But, the seductive statement that she "receives no new ... information" is false. In earlier posts it has been shown that when she is awakened she has different information from at the earlier stage. Not least, that now it might be Tuesday.

But, the halfer argument denies that that information is valid for the purposes of calculating probabilities. And, further, that the definition of probability as the limit of relative frequency cannot be used in this case.

There are many posts in this thread trying to track down why relative frequencies cannot be used in this case.

That, among other things, is what the argument is about:

Why can't the sleeper argue: "if it is Tuesday ..."?

And, why can't the sleeper use the hypothetical limit of relative frequencies?

The halfer argument is, essentially, that because these techiques are invalid in this particular case, then the only way to calculate the probability is to adopt the a priori probability of 1/2.

That's one aspect of the debate. There's also been a debate about why the Bayesian approach can/cannot be used. The halfer position depends on Bayes being outlawed (in this case) as well.

 

[PeterDonis]

the halfer argument denies that that information is valid for the purposes of calculating probabilities.

No, it doesn't. It just denies that the probability you calculate using that information is the "subjective credence that the coin came up heads". Which is vague ordinary language. If Beauty was asked the question "what is the conditional probability that the coin came up heads, given that this particular time at which you are awakened is randomly chosen from three equally likely possibilities?" then there would be no debate. But that isn't the way the question is asked in the standard version of the problem.

 

[PeroK]

No, it doesn't. It just denies that the probability you calculate using that information is the "subjective credence that the coin came up heads". Which is vague ordinary language. If Beauty was asked the question "what is the conditional probability that the coin came up heads, given that this particular time at which you are awakened is randomly chosen from three equally likely possibilities?" then there would be no debate. But that isn't the way the question is asked in the standard version of the problem.

That's a good point. So, I have only two questions:

Why is my subjective credence not the probability that I can calculate?

Why is the subjective credence that a coin is heads 1/2 in the first place? If it's not a calculated probability, then where does it come from? If a coin is too simple, then we could take an example where it takes probability theory to come up with the basic number in the first place.

My position is that if you have to give a numerical value to subjective credence (and that is the only possible value), then it must be 1/3. In other words, 1/3 cannot be wrong.

Moreover, if 1/2 is a valid answer, then so is 1 - the subjective certainty (that a gambler may have) that the coin must be heads.

In summary, the halfers are using probability theory selectively (to get 1/2 in the first place), then denying it to get their subjective credence - and that is not self-consistent (*).

Probability theory, on the other hand, is effectively self-consistent and can be used throughout.

(*) Although many halfers on this thread, actually state their answer as a probability.

 

[PeterDonis]

Read the Wikipedia page

Since you referenced this page, I'll go ahead and critique another aspect of the thirder argument as it's presented there:

"Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, her credence that it is Monday should equal her credence that it is Tuesday since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus

P(Tails and Tuesday) = P(Tails and Monday).

Consider now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. She knows the experimental procedure doesn't require the coin to actually be tossed until Tuesday morning, as the result only affects what happens after the Monday interview. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should therefore hold that P(Tails | Monday) = P(Heads | Monday), and thus

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday)."

This argument is based on supposing that Beauty has information that she doesn't actually have, and the information is different in the two cases. In other words, first it is argued that the following two conditional probabilities are equal: P(Monday|Tails) = P(Tuesday|Tails). In other words, if Beauty knows the coin came up tails, it is equally probable that she was awakened on either day.

Second, it is argued that the following two conditional probabilities are equal: P(Tails|Monday) = P(Heads|Monday). In other words, if Beauty knows that it is Monday, it is equally probable that the coin came up heads or tails.

Now that I've correctly stated the actual conditional probabilities, it should be obvious that the third stage of the argument is invalid, since it is arguing, in essence, that P(Monday|Tails) = P(Tails|Monday). But that is only true if we fill in the numerical values in the two conditional probabilities above (using the constraint that they must add up to 1), and hence observe that P(Monday|Tails) = P(Tails|Monday) = 1/2. But the argument purports to show that P(Tails and Monday), which is how it describes both of the conditional probabilities P(Monday|Tails) and P(Tails|Monday), is 1/3. This is obviously false.

Here's another argument based on conditional probabilities. First we need to find an expression for a conditional probability of Heads that is conditioned on the information Beauty actually has. This is P(Heads|Awakened), where "Awakened" means Beauty has just been awakened but doesn't know which day, Monday or Tuesday, she has been awakened in. Then the obvious way to proceed is to write:

P(Heads|Awakened) = P(Heads|Monday) P(Monday|Awakened) + P(Heads|Tuesday) P(Tuesday|Awakened)

Since the conditions of the experiment tell us that P(Heads|Tuesday) = 0, and we know from the above that P(Heads|Monday) = 1/2, we now have only to evaluate P(Monday|Awakened). I would expect a thirder to claim that P(Monday|Awakened) = 2/3, by arguing that there are three possible "awakenings", Monday & Heads, Monday & Tails, and Tuesday & Tails, and that these are all equally probable. But a halfer could argue that when Beauty is awakened, it isn't a random choice between those three alternatives. She gets awakened on Monday, and then the coin flip result is checked to see if she is awakened again on Tuesday. And then we get into all the arguments about relative frequencies and Bayesian priors and so on. But notice that, if we are arguing about that stuff, that means we have reinterpreted Beauty's "subjective credence that the coin came up heads" as the conditional probability P(Heads|Awakened), which, as far as I can tell from the Wikipedia page, nobody has even brought up.

 

[PeterDonis]

Why is my subjective credence not the probability that I can calculate?

It's a probability you can calculate. But which one? There are always many probabilities you could calculate. Which one is the one the vague ordinary language is asking for?

Why is the subjective credence that a coin is heads 1/2 in the first place?

I can't believe you're seriously asking this, but my answer would be that this is my Bayesian prior based on the assumption that it's a fair coin. (The conditions of the experiment seem to assume that it's a fair coin, and everyone who talks about it also seems to assume that.) If you would rather just deal with conditional probabilities, I would say it is the obvious assumption that everyone is making for P(Heads|Flipped).

if you have to give a numerical value to subjective credence (and that is the only possible value), then it must be 1/3. In other words, 1/3 cannot be wrong.

These two statements are not logically equivalent. The first states that 1/3 must be right. The second states that 1/3 cannot be wrong. But "cannot be wrong" only equates to "must be right" if there is only one possible right answer. And that is only true if the problem is stated using precise math, not vague ordinary language. I notice that in repeated responses you have not once addressed the issue of the ordinary language being vague.

the halfers are using probability theory selectively (to get 1/2 in the first place), then denying it to get their subjective credence

I still don't see this at all. The conditional probability P(Heads|Flipped) is 1/2--everyone appears to agree on that. The halfers are simply saying that this conditional probability is the one that corresponds to the vague ordinary language "subjective credence that the coin landed heads". That's all there is to it. Nobody is denying probability theory. They're just picking a different conditional probability than you are. Once again, in repeated responses you have failed to address this issue at all. You appear to believe that ordinary language can never be vague, which seems absurd to me.

 

[PeroK]

Since you referenced this page, I'll go ahead and critique another aspect of the thirder argument as it's presented there:

"Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, her credence that it is Monday should equal her credence that it is Tuesday since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus

P(Tails and Tuesday) = P(Tails and Monday).

Consider now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. She knows the experimental procedure doesn't require the coin to actually be tossed until Tuesday morning, as the result only affects what happens after the Monday interview. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should therefore hold that P(Tails | Monday) = P(Heads | Monday), and thus

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday)."

This argument is based on supposing that Beauty has information that she doesn't actually have, and the information is different in the two cases. In other words, first it is argued that the following two conditional probabilities are equal: P(Monday|Tails) = P(Tuesday|Tails). In other words, if Beauty knows the coin came up tails, it is equally probable that she was awakened on either day.

Second, it is argued that the following two conditional probabilities are equal: P(Tails|Monday) = P(Heads|Monday). In other words, if Beauty knows that it is Monday, it is equally probable that the coin came up heads or tails.

Now that I've correctly stated the actual conditional probabilities, it should be obvious that the third stage of the argument is invalid, since it is arguing, in essence, that P(Monday|Tails) = P(Tails|Monday). But that is only true if we fill in the numerical values in the two conditional probabilities above (using the constraint that they must add up to 1), and hence observe that P(Monday|Tails) = P(Tails|Monday) = 1/2. But the argument purports to show that P(Tails and Monday), which is how it describes both of the conditional probabilities P(Monday|Tails) and P(Tails|Monday), is 1/3. This is obviously false.

Here's another argument based on conditional probabilities. First we need to find an expression for a conditional probability of Heads that is conditioned on the information Beauty actually has. This is P(Heads|Awakened), where "Awakened" means Beauty has just been awakened but doesn't know which day, Monday or Tuesday, she has been awakened in. Then the obvious way to proceed is to write:

P(Heads|Awakened) = P(Heads|Monday) P(Monday|Awakened) + P(Heads|Tuesday) P(Tuesday|Awakened)

Since the conditions of the experiment tell us that P(Heads|Tuesday) = 0, and we know from the above that P(Heads|Monday) = 1/2, we now have only to evaluate P(Monday|Awakened). I would expect a thirder to claim that P(Monday|Awakened) = 2/3, by arguing that there are three possible "awakenings", Monday & Heads, Monday & Tails, and Tuesday & Tails, and that these are all equally probable. But a halfer could argue that when Beauty is awakened, it isn't a random choice between those three alternatives. She gets awakened on Monday, and then the coin flip result is checked to see if she is awakened again on Tuesday. And then we get into all the arguments about relative frequencies and Bayesian priors and so on. But notice that, if we are arguing about that stuff, that means we have reinterpreted Beauty's "subjective credence that the coin came up heads" as the conditional probability P(Heads|Awakened), which, as far as I can tell from the Wikipedia page, nobody has even brought up.

I'm not sure I follow all those arguments. The basic conditional probability calculations should all be consistent. If not, then I fail to see what it is about this particular problem that makes them inconsistent.

For example, in an earlier post we showed that Beauty would agree with a random observer who happened on the experiment - they could discuss and agree that they had precisely the same information about the problem.

Therefore, any argument that applies to her must equally apply to the random observer. And, if the random observer concludes that the probablity of heads is 1/2, then bang goes all probability theory. Or, at least all conditional probability theory. And, we are back to the probability being 1/2 until we know for sure that it's heads or tails.

On the final point. If the subjective credence is not P(Heads|Awakened) - i.e. a conditional probablity given that she has been awakened, then what is the point of the experiment? If the subjective credence is simply P(Heads), then there is no need for an experiment. Then, the answer to any question in any experiment is simply 1/2. P(Heads) remains 1/2 and doesn't change (until you look at the coin, I guess).

Perhaps that is the root of the issue. The halfers don't see a problem with specifying a complicated experiment and having someone in the middle of the experiment say that their subjective credence does not depend on where they are in the experiment, but only on a pre-experiment value.

 

[stevendaryl]

I still don't see this at all. The conditional probability P(Heads|Flipped) is 1/2--everyone appears to agree on that. The halfers are simply saying that this conditional probability is the one that corresponds to the vague ordinary language "subjective credence that the coin landed heads".

I think the disagreement is about whether (and how) to update the a-priori probability of heads in light of new information. Certainly, if I told you that I flipped a second coin, and at least one of the two result was "heads", then your subjective credence that the first coin was heads would change. (I think it would change from 1/2 to 2/3). So usually, "subjective credence" includes whatever information is available.

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.

 

[PeroK]

I think the disagreement is about whether (and how) to update the a-priori probability of heads in light of new information. Certainly, if I told you that I flipped a second coin, and at least one of the two result was "heads", then your subjective credence that the first coin was heads would change. (I think it would change from 1/2 to 2/3). So usually, "subjective credence" includes whatever information is available.

Yes, and in fact most of Peter's objections would appear to apply to any regular problem. Which probability do you calculate in any case? The only factor that makes this problem different is the amnesia drug. I don't see the argument that blows the conditional probability argument in the case of an amnesia drug, that doesn't also blow the conditional probability argument in any regular problem.

Anyway, it's getting late for me and I can't summon the strength to argue any longer.