Undergrad The Sleeping Beauty Problem: Any halfers here?

[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.

 

[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.