The Sleeping Beauty Problem: Any halfers here?

[Dale]

My point was that both A and B are consistent with the description of the scenario as given in the Wikipedia article linked to in the OP of this thread;

This is where we disagree. I think that B is not consistent with the description.

I understand that this is how you are defining "credence". I am just pointing out that this is a definition.

This is not a personal definition of mine, this is the standard definition. Credence is used as a synonym of subjective probability. Operationally, when a person is asked for their credence for X they are being asked for the betting odds at which they would be indifferent to a wager for X (then changed from odds form into probability form). Your scenario B is not a wager for X, it is a wager for X and Y.

I don't think "credence", or probability for that matter, is an intrinsic property. It depends on what purpose you are going to use it for.

I think that you are introducing a personal definition here. Credence is used to determine if a person will accept or reject an offered wager at a given price. The confusion here only comes because you want to offer a different wager.

But that, in itself, is an argument about what words should mean,

I recognize that, but you are claiming that the problem is insufficiently specified, and your justification for that is to use a nonstandard meaning for credence and show that with your non standard meaning of credence multiple outcomes are possible.

If the Sleeping Beauty problem were posed as part of a homework assignment then part of the assignment would be to test the student's understanding of the standard meaning of the important terms in the problem. A student giving an answer of 1/2 would be wrong, even if the reason they are wrong is because they misunderstood the meaning of credence.

 

[stevendaryl]

Consider "the 1001 beauties."

On sunday, 1 beauty is randomly selected as the winner. The other 1000 lose. The winner wakes up every day for 1000 days. The losers each wake up on one day in random order.

Using your argument, when they wake up each beauty should have 1/2 credence that they are the winner. And that seems plausible since they wake up next to another beauty who seemed to begin with the same information, and therefore should divide the probability evenly between them.

But there is first-person information which cannot be shared. If we were to select a random beauty, then randomly select someone who wakes up next to them, the chances would be 1000/1001 that the person waking up next to them is the winner. If we were to select a random day, and then ask the probability of each beauty on that day being the winner, it would be 1/2. But I believe that each beauty can begin by reasoning that "I am randomly selected from the beauties", "I am awake today as a result of a process of being randomly selected from the beauties and then having the day(s) of awakening randomly chosen", information which is first-person, related to the process of discovering information, and can't be shared. If I believe that I am randomly selected from the beauties I can't believe that a beauty waking up next to me is also randomly selected from all the beauties. After all, I was almost certain I'd be waking up next to the winner.

It's a strange situation, definitely. You have two people who are awake. One of them is the winner. One is one of the 1000 losers. By symmetry, they can't come to different conclusions about their chances of being the winner since they have exactly the same information.

The analogy of the thirder argument is:

"One of us is the winner. There is no indication of which one. So I might as well assume that it's a 50/50 chance that it's me."

The analogy of the halfer argument is:

"I only had a 1/1000 chance to start with. I knew ahead of time that I would eventually be awake at the same time as the winner. So there is really no reason to think that I'm more likely to be the winner now."

In the first case, both come to the conclusion that they are likely to be winners. In the second case, both come to the conclusion that it is very unlikely that they are the winner (even though they know that one of them is very wrong).

I really do understand the intuitions behind the halfer position in this case.

I suddenly realized that there is a connection with nonmeasurable sets. Assuming the continuum hypothesis, it is possible to have an ordering $\prec$ on the reals in the range $[0,1]$ such that for every $x$, there are countably many values of $y$ such that $y \prec x$, but uncountably many $y$ such that $x \prec y$.

So suppose Alice and Bob each generate a random number in the range $[0,1]$. Call Alice's number $x$ and Bob's number $y$. Now, ask Alice what are her subjective odds that $y \prec x$. She has two ways of reasoning about this:

1. On the one hand, she knows that the situation is exactly symmetrical. So the odds should be 50/50.
2. On the other hand, when she looks at $x$, she knows that there are only countably many values of $y$ such that $y \prec x$. So the odds that Bob picked one of those at random is 0. (Any countable set has measure zero)

So depending on how she looks at it, her odds of having the largest number (according the ordering $\prec$) is either 50/50 or 0.

 

[Stephen Tashi]

If the Sleeping Beauty problem were posed as part of a homework assignment then part of the assignment would be to test the student's understanding of the standard meaning of the important terms in the problem. A student giving an answer of 1/2 would be wrong,

They'd be wrong if you were the grader.

even if the reason they are wrong is because they misunderstood the meaning of credence.

Are you saying that there is no ambiguity about how credence is defined in the Sleeping Beauty problem?

The Wikipedia, (which might be incorrect), says:

Therefore, the Sleeping Beauty problem is not about mathematical probability theory. Rather, the question is whether subjective probability or credence are well-defined concepts, and how they must be operationalized.

The problem has apparently been the subject of many published articles. Can we appeal to the "weight of authority" and find there is a consensus answer in 2017? The thirders seem to publish the majority of articles, but the problem still appears to be controversial.

 

[PeterDonis]

That puts the first bet (in the case of two) in a strange position: You want to define probability in terms of consequences, but the first bet has no consequences.

"No consequences" (zero payoff regardless of the bet) is still "consequences".

Of course, the person making the bet doesn't know that it has no consequences...

Yes, and that would be a key factor in the rationale for that betting scheme if I were the one setting it up. It would be something like: we don't want to "privilege" the situation where Beauty gets two chances to bet instead of one, because both bets must be the same anyway, since Beauty has the same information when making both of them. So we only count the last one.

Yes, this might seem "strange", but so is the whole scenario. Such strange scenarios are typical of philosophical thought experiments, so I don't think "hey, that's strange" is a valid reason to discard an alternative. The point of the strangeness is to push ideas to their limits to see what happens.

 

[PeterDonis]

Your scenario B is not a wager for X, it is a wager for X and Y.

See, here's that vague ordinary language again. I say scenario B is a wager for X, but with different payoffs than scenario A. In both scenarios, Beauty is asked for her credence that the coin came up heads. She is not asked in scenario B for her credence that the coin came up heads and it is Monday. She is asked the same question in both scenarios. Same question, same wager. But prior to the experiment starting, in A and B, she is told different payoff schemes, and that affects her answer to the question when she is asked during the experiment.

you are claiming that the problem is insufficiently specified, and your justification for that is to use a nonstandard meaning for credence

No, I'm using the standard meaning for credence, and pointing out that a wager is insufficiently specified if no payoffs are given. The original scenario does not give payoffs.

A student giving an answer of 1/2 would be wrong, even if the reason they are wrong is because they misunderstood the meaning of credence.

What if the student pointed out to you, the professor, that the standard definition of credence, which you gave in class, involves a wager, and a wager is insufficiently specified if there are no payoffs given, and your exam question gives no payoffs?

 

[stevendaryl]

"No consequences" (zero payoff regardless of the bet) is still "consequences".

Well, it's a degenerate case in which you have no possibility of losing and no possibility of winning. How do you define the probability in a case like that? The betting definition of probability is something like: The probability of winning is the amount (in dollars) I would be willing to bet in order to have a chance to win one dollar. If it's impossible to win or lose any money, then the probability would be undefined. It wouldn't make any difference if I bet 1/100 or 1.

Yes, and that would be a key factor in the rationale for that betting scheme if I were the one setting it up. It would be something like: we don't want to "privilege" the situation where Beauty gets two chances to bet instead of one, because both bets must be the same anyway, since Beauty has the same information when making both of them. So we only count the last one.

Okay, well, that's equivalent to treating the two-wakenings as the same as the one-wakening, which is basically ignoring all the details about wakenings and memory wipes, etc. I guess that's okay, but it seems contrary to the spirit of the puzzle.

 

[PeterDonis]

it's a degenerate case in which you have no possibility of losing and no possibility of winning. How do you define the probability in a case like that?

Ah, ok, I see the issue you're raising, but I think the answer is that this case is an edge case in the ordinary language term "wager". I would say that the actual "wager" is made on Sunday, when Beauty is told the experimental protocol, including the payoff scheme, and has to decide how she will respond when she is awakened and asked what is her subjective credence that the coin came up heads. (She can decide in advance because she knows exactly what information she will have on each awakening.) And what she decides on Sunday will depend on what payoff scheme she is told. So however many awakenings and askings happen during the experiment, they are all part of one "wager", which gets resolved on Wednesday, when the experiment concludes.

Beauty has enough information on Sunday to compute correct expectations for whatever payoff scheme she is told, so in that sense she has all of the "probability" information she needs. If the coin comes up tails, it is true that the actual answer she gives on Monday will not affect her actual winnings on Wednesday at all, but that fact is irrelevant to her decision of what to answer, since her answer is precomputed in advance, and will be the same on both Monday and Tuesday.

I agree this is a somewhat unusual use of the term "wager", but as I said before, the whole scenario is unusual. We don't normally think of it being possible for a person to be offered multiple "wagers" in each of which they have exactly the same information (where "information" here includes all of their memories).

 

[JeffJo]

Consider "the 1001 beauties."

On sunday, 1 beauty is randomly selected as the winner. The other 1000 lose. The winner wakes up every day for 1000 days. The losers each wake up on one day in random order.

Using your argument, when they wake up each beauty should have 1/2 credence that they are the winner.

And using yours, you seem to think it depends on the number 1001.

What if the number of contestants is unknown to them (except that N>=2); each just finds herself awake in a room with another Beauty, with no recollection of prior days. Just the process that requires one of them to be the winner, and the other to be one of however many losers. Each has the same information, so their credences can't be different. If they must sum to 1, they are each 1/2 regardless of the number. If they don't have to sum to 1, what are they? Do they need to know N? Is there even a mathematics that can tell you what they should sum to?

Or is it simply 1/2? That is an incredibly obvious answer: no matter what circumstances led to each of you being awake at the moment, no matter how different they are, the fact is that your observation is only that one of you was selected by process A, one by process B, and you have no credence that A is preferred over B, or B over A.

Or try this, in the same vein: you win a one-in-a-million lottery for a free cruise. You get to pick 10 friends, one of whom will be selected at random to accompany you. The only stipulation is that the two of you will be amnesia-ed the first night, so that neither knows which won the lottery, or was named as a friend. Does your credence that you won the lottery depend on the number 1,000,000, or 10? Or both? How about your credence that you are the friend? Does either change if you get to go on 5 different cruises, each with a different friend from your list but no memory of the other cruises?

Or is it simply 1/2?

You are the one who is trying to make information, that is available only in the global sense, part of the solution to a local-memory-only question.

 

[Dale]

I say scenario B is a wager for X, but with different payoffs than scenario A.

Scenario B doesn't just change the amount that is paid for a win, it changes the conditions under which Beauty wins. In scenario A the wager can be resolved by revealing the coin to Beauty. In scenario B resolving the wager requires revealing not only the coin but also a clock/calendar.

 

[PeterDonis]

Scenario B doesn't just change the amount that is paid for a win, it changes the conditions under which Beauty wins.

The conditions you refer to are not specified in the original specification of the problem, so even under this interpretation I don't see how scenario B is inconsistent with the original specification of the problem.

At this point it looks to me like we're disagreeing about how to interpret vague ordinary language, which was the original point I was trying to make in this thread: that ordinary language is vague.

In scenario B resolving the wager requires revealing not only the coin but also a clock/calendar.

I'm not sure I agree with this, because (a) the coin result determines the clock/calendar possibilities, and (b) Beauty gives the same answer each time she is asked during the experiment anyway (since she has the same information at each awakening), so the clock/calendar state each time she give the answer doesn't actually affect anything; the coin flip result (plus her precomputation of the answer she will give) is already sufficient to determine everything else.

To put it another way: on Wednesday, when the experiment ends, what information does the experimenter have to give Beauty to resolve all bets? Does he have to show her the clock/calendar states for each of the times she was awakened and asked her credence? No. All he has to show her is the coin.

Again, I think this is just exposing edge cases in ordinary language terms that we usually don't bother digging into in this much detail, because our normal usage of those terms doesn't give rise to those edge cases. We only see the edge cases in specially constructed thought experiments.

 

[Dale]

Are you saying that there is no ambiguity about how credence is defined in the Sleeping Beauty problem?

The only ambiguity I can see is whether or not Beauty is rational in her assessment of subjective probability and if the experimenters are non-malicious. If they are, then the rest follows unambiguously.

 

[Dale]

I don't see how scenario B is inconsistent with the original specification of the problem.

It is inconsistent with the specification because Beauty is asked for her current credence that the coin toss was heads. That specifies a wager which is resolved simply by revealing the outcome of the coin toss, and she is asked to name the odds at which she would be ambivalent.

Your scenario B would be something like her current credence that the coin toss was heads and today is Monday. Resolving such a wager would require revealing both the coin toss and the time. It is a different credence.

At this point it looks to me like we're disagreeing about how to interpret vague ordinary language, which was the original point I was trying to make in this thread: that ordinary language is vague.

To me this seems more like one of the cases where the technical meaning is not being correctly understood and applied. This sort of thing happens often in the relativity forum, such as when we say "proper time" and the people reading understand it wrong.

the clock/calendar state each time she give the answer doesn't actually affect anything

It certainly affects whether or not she would win the current hypothetical wager in your scenario B.

To put it another way: on Wednesday, when the experiment ends, ...

She is specifically being asked to compute her current credence, not her credence on Wednesday. Again, this is unambiguously excluded by the description.

 

[PeterDonis]

Beauty is asked for her current credence that the coin toss was heads. That specifies a wager which is resolved simply by revealing the outcome of the coin toss

I understand that this is your interpretation of the original specification of the problem in the Wikipedia article. But I don't think it's the only possible interpretation, since the original specification says nothing about any wager at all. If there is some unique "default wager" that is implied by any use of the term "credence" that doesn't include an explicit specification of a wager, I would like to see some actual references from the literature on this subject that show that. As I said some posts back, my knowledge of that literature is not extensive. But from what I do know of it I don't think the definition of the term "credence" is pinned down to that extent.

To me this seems more like one of the cases where the technical meaning is not being correctly understood and applied.

That is why I have asked for a reference in the literature that explains that technical meaning. If I'm mistaken as to how much of a consensus there is on the term "credence" and its meaning, I would like to know it.

It certainly affects whether or not she would win the current hypothetical wager in your scenario B.

No, it doesn't. I explained why not. It's the same kind of point that you made about reformulating the problem. You are using a formulation in which you have to know the clock/calendar state each time Beauty is awakened in order to resolve wager B. I am giving an alternative formulation in which you don't: all you need to know is how the coin flip came out, plus the fact that Beauty's answer each time she is awakened will be the same. (And on Wednesday, you don't even need to deduce from the experimental protocol that Beauty's answer each time was the same; you know it because it has already happened that way.) Since her answer each time is the same, you don't have to know the clock/calendar state for any of the answers, since it doesn't matter--it drops out of the calculation.

 

[Dale]

If there is some unique "default wager" that is implied by any use of the term "credence" that doesn't include an explicit specification of a wager, I would like to see some actual references from the literature on this subject that show that.

I will look for that. The credence of X is the subjective probability that X is true. I will find some references that mention that and look for some that discuss the idea of using hypothetical wagers to identify the credence.

As it applies here, the problem with your assessment is that you are not sticking to the requested probability. You are asking the credence on Wednesday and/or the credence of heads + Monday

I am giving an alternative formulation in which you don't: all you need to know is how the coin flip came out, plus the fact that Beauty's answer each time she is awakened will be the same.

That is not all that you need to know to calculate the outcome of each individual wager. Beauty is asked to give her credence (potentially) multiple times. If it is tails then in your scenario B if she bet tails she will win one and draw one. How do we know which wager she won and which she drew? We have to know which one was on Monday and which was on Tuesday. So the outcome of the individual wager depends on more than just the result of the coin toss, so it is not just the credence of the result of the coin toss.

Suppose you personally (not Sleeping Beauty) were at a horse race and with perfect knowledge of horse race odds you wanted to bet Galloping Gertie to win. But at this track when you go to collect your winnings they pull out a deck of cards and if they drew a spade the bet would be a draw. Would you honestly say that the odds you would be indifferent to are the same as the odds of Galloping Gertie actually winning?

 

[PeterDonis]

If it is tails then in your scenario B if she bet tails she will win one and draw one. How do we know which wager she won and which she drew? We have to know which one was on Monday and which was on Tuesday.

No, we don't, because we already know in advance that she will give the same answer both times. And that means it doesn't matter which one was on Monday and which one was on Tuesday. That would only matter if the Monday answer was different (or could have been different) from the Tuesday answer, and we know it isn't (and couldn't).

 

[PeterDonis]

Suppose you personally (not Sleeping Beauty) were at a horse race and with perfect knowledge of horse race odds you wanted to bet Galloping Gertie to win. But at this track when you go to collect your winnings they pull out a deck of cards and if they drew a spade the bet would be a draw. Would you honestly say that the odds you would be indifferent to are the same as the odds of Galloping Gertie actually winning?

No. But now let's pursue your implied analogy by trying to modify this scenario, which I'll call the Galloping Gertie (GG) scenario, to reproduce the key features of the Sleeping Beauty (SB) scenario, including version B of post #67, that I've been focusing on. There are three such features:

First, in the SB scenario, the event whose odds I have perfect prior knowledge of (the result of the coin flip, isolated from anything else) is also the event that determines which, if any, of my bets will be a draw instead of being paid.

Second, in the SB scenario, one of my bets is always not drawn regardless of the outcome of the coin flip; there is no coin flip result which makes all of my bets draws.

Third, in the SB scenario, the coin flip is made before I am asked to make any bet that might possibly turn out to be a draw.

My modified GG scenario that incorporates these features looks like this: after having the experimental protocol explained to me, I get put in a stasis box (from Larry Niven's sci-fi novels--basically it isolates me completely from the rest of the universe and keeps me in the same state, no passage of time for me, indefinitely) while the race is being run. If GG wins, I am taken out of the stasis box and allowed to place my bet before being told of the result of the race; then my bet is honored. If GG loses, I am taken out of the box and allowed to place my bet without being told the result of the race; then, before that bet is resolved, I am given an amnesia drug that makes me forget everything that happened after I was taken out of the box; then I am allowed to place my bet again without being told the result of the race. Only the second bet is honored; the first is drawn.

Of course this scenario is outlandish; but so is the SB scenario (even in the original version without the post #67 additions).

 

[Dale]

My modified GG scenario that incorporates these features looks like this: after having the experimental protocol explained to me, I get put in a stasis box (from Larry Niven's sci-fi novels--basically it isolates me completely from the rest of the universe and keeps me in the same state, no passage of time for me, indefinitely) while the race is being run. If GG wins, I am taken out of the stasis box and allowed to place my bet before being told of the result of the race; then my bet is honored. If GG loses, I am taken out of the box and allowed to place my bet without being told the result of the race; then, before that bet is resolved, I am given an amnesia drug that makes me forget everything that happened after I was taken out of the box; then I am allowed to place my bet again without being told the result of the race. Only the second bet is honored; the first is drawn.

And as a result of those wagers what then is your credence that Galloping Gertie will win?

 

[Dale]

No. But now ...

I submit to you that the argument over the meaning of credence is over. You recognize that a wager on A with conditions to turn the wager to a draw is no longer about the subjective probability of A but about some probability of A and the conditions.

 

[PeterDonis]

And as a result of those wagers what then is your credence that Galloping Gertie will win?

The same as if I were just betting normally.

You recognize that a wager on A with conditions to turn the wager to a draw is no longer about the subjective probability of A but about some probability of A and the conditions.

My point is that, in the SB scenario, the "conditions to turn the wager to a draw" (the result of the coin flip) are the same as "A", the thing being wagered on (the result of the coin flip). In your GG scenario before my modifications, that is not true; the conditions (drawing a spade or not) are different from the thing being wagered on (GG winning).

If you want to say that that makes this a discussion about "conditions" rather than about the meaning of "credence", that's fine. Then my response is that the Sleeping Beauty scenario as described on the Wikipedia page is not a good scenario for distinguishing "credence" from "conditions" that are not included in the concept of credence, because of the property described in the previous paragraph. Your GG scenario (before my modifications) would be a much better example to distinguish those concepts.

 

[forcefield]

the standard definition of credence... involves a wager

That definition does not make sense in the context of the voting options. Of course you can always add voting options but note that a question does not have a meaning independent of the voting options.

Also I'd like to add that some pages ago when I said "awake", I could as well have said "awakened", and yet the conclusion would have remained. Whatever it was.

 

[Dale]

the "conditions to turn the wager to a draw"

In all of my reading on credence and subjective probability I have never once seen the hypothetical wager be one where a draw is a possibility. Frankly, it is completely bizarre.

 

[stevendaryl]

To me, the definitive way of highlighting the two ways of reasoning is the similar paradox inspired by @JeffJo and @Marana:

We have a contest (a lottery) with 1001 contestants. Only one of them wins the big prize. The strange rules of the contest say the following:

• If you play, you will not be told whether you win for 1000 days.
• Each loser will be awakened on a different day, and will sleep through the other 999 days.
• The one winner will be awakened each of the 1000 days.
• Through the use of drugs and alcohol, none of the contestants will know what day it is, nor whether they've ever been awakened previously.

With this arrangement, there will be two contestants awake each of the 1000 days: One loser, and the winner. They meet and try to figure out who is the winner and who is the loser. The two kinds of reasoning can be summarized as follows:

1. Halfer reasoning: "I knew before the first day that I only had a 1/1001 chance of winning the contest. There is no reason to think differently now. So I still have a 1/1001 chance that I am the winner."
2. Thirder reasoning: "One of us is the winner. There is no basis for thinking it's more likely to be you or me. So I conclude that there is 50% chance that I'm the winner."

Both responses sound eminently reasonable to me. But from a betting notion of probability, the halfer reasoning is a little strange: Each person is willing to bet that the other one is the winner. More precisely, if both people are halfers, then they should each be willing to say: "If I'm the winner, I'll pay you $1001. If you're the winner, you pay me $1". But if they each place such bets at every opportunity, then the winner will end up paying out over $1 million. (I hope the prize money for the contest covers that.)

 

[JeffJo]

Halfer reasoning: "I knew before the first day that I only had a 1/1001 chance of winning the contest. There is no reason to think differently now. So I still have a 1/1001 chance that I am the winner."

Continued halfer reasoning: But had I met you, fellow contestant, before the winner was determined? I would have also said you had a 1/1001 chance. The problem is that both of these are prior probabilities. The issue is, how should we update them? Or can we - are the posterior probabilities the same as the priors?

If there is "new information" here, then I don't actually need to know how to define it in order to know what answer is correct. Because I know the result of however it applies to Bayes Theorem has to result in the expression (1/1001)/[(1/1001)+(1/1001)]=1/2.

But if there is no new information, and the probability I won is still 1/1001, then the probability, from my perspective, that you won is 1000/1001. Because there is a 100% chance that one of us has won. Since this represents an update from your prior probability of 1/1001, there must be new information.

Each person is willing to bet ...

I refuse to consider betting arguments. Even if they get the right answer, they are invalid since the total amount risked - the $1 million you spoke of - is a function of the outcome of the bet.

 

[stevendaryl]

I refuse to consider betting arguments. Even if they get the right answer, they are invalid since the total amount risked - the $1 million you spoke of - is a function of the outcome of the bet.

Well, the whole point of betting is that it's a way of making sense of subjective probabilities.

 

[JeffJo]

Well, the whole point of betting is that it's a way of making sense of subjective probabilities.

And the point of avoiding them, is that there has never been a person on either side of the halfer/thirder fence who was swayed to the other side by one. They will just claim the bet must be made within the perspective they think should apply.

 

[stevendaryl]

And the point of avoiding them, is that there has never been a person on either side of the halfer/thirder fence who was swayed to the other side by one. They will just claim the bet must be made within the perspective they think should apply.

On the contrary, when you make it into a bet, everyone agrees on how you should bet.

 

[stevendaryl]

And the point of avoiding them, is that there has never been a person on either side of the halfer/thirder fence who was swayed to the other side by one. They will just claim the bet must be made within the perspective they think should apply.

It's actually not true that nobody's opinion has been changed by the discussion. I started out slightly favoring the halfer position, but now I'm pretty solidly a thirder.

 

[JeffJo]

I started out slightly favoring the halfer position, but now I'm pretty solidly a thirder.

Then you were on the fence leaning one way, and are now just stepping off onto the thirder side. Sorry if I was unclear, but I meant people who had stepped off completely. :)

My point is that once people decide on an answer, they force their opinions about how to apply perspective in order to achieve that answer. Since any betting argument has to start with a perspective, and the committed opinions have already assumed their perspective can't be wrong, a betting argument in a different perspective won't sway them in the slightest. What I'm trying to do, is to convince people, outside of perspective, that the answer can only be 1/3. So maybe their perspective needs to change.

 

[Marana]

I don't see much symmetry between the heads and tails situations. How do you get 1/2 from this situation?

Sleeping Beauty could reason, as in the original situation:

• Under the assumption that today is Monday, I would conclude that there is an equal probability of heads and tails. So P(Heads & Monday) = P(Tails & Monday)
• Under the assumption that the coin toss result was tails, I would conclude that there is an equal probability of it being Monday or Tuesday. So P(Tails & Monday) = P(Tails & Tuesday)

Those two assumptions imply that the three probabilities are equal: P(Heads & Monday) = P(Tails & Monday) = P(Tails & Tuesday)

(By "it is monday", I mean that according to nondilated clocks.)

I'm not sure I understand how the time dilation affects the answer.

How are you getting P(Heads & Monday) = P(Tails & Monday) in this version? The coin can't be flipped on monday now (and remember it is flipped on sunday in the original) because the time dilation drug must be given before waking monday.

You're right that it isn't really symmetric... only in the sense that at literally every time both beauties are awake and experiencing something which doesn't distinguish between heads and tails.

But this version makes it clear that you need to explain how you are working with time. It is plausible that you could model time by saying that you should think of yourself as randomly selected from, say, all 1-second intervals. But that would support the halfer answer in this version since there are equally as many 1-second intervals on heads and tails.

That's just one possibility, and you may well have a better way to do it, but it needs to be explained.

Or try this, in the same vein: you win a one-in-a-million lottery for a free cruise. You get to pick 10 friends, one of whom will be selected at random to accompany you. The only stipulation is that the two of you will be amnesia-ed the first night, so that neither knows which won the lottery, or was named as a friend. Does your credence that you won the lottery depend on the number 1,000,000, or 10? Or both? How about your credence that you are the friend? Does either change if you get to go on 5 different cruises, each with a different friend from your list but no memory of the other cruises?

There are some extra factors in your version. If I discover that "Andy" is my shipmate, I may know that he isn't in my top 10 friends and therefore he is the winner with probability 1. I may not be indifferent about who among my friends is likely to play the lottery.

But I can see it's intended to be the 1001 beauties situation. Yes, the number of beauties matters, and it's because there is an experiment. On sunday you are told the process that determines everything that follows. You only have a 1/1001 chance of winning, and a 1000/1001 chance of waking up next to the winner. Drawing the conclusion that you are the winner with probability 1/2 after waking up is essentially oversampling, because you can never learn "I am asleep". After correcting for the oversampling, the probability that you are the winner is 1/1001.

Keep in mind, this argument only holds when you retain your memory of the sunday experiment. "Repeating the experiment" requires repeated sundays, which you can always remember, and which result in the 1/1001 answer. If you wake up and are only then told "by the way, there was an experiment on sunday..." you can't use this argument. In such a situation you would have to ask yourself "why did I wake up in this experiment?", a question which can lead to the thirder answer with some stipulations. There must be an unbroken line of causation from the sunday experiment to your waking up.

I think that you are introducing a personal definition here. Credence is used to determine if a person will accept or reject an offered wager at a given price. The confusion here only comes because you want to offer a different wager.

Out of curiosity, in the lottery version (suppose on sunday you buy a 1 in a million lottery ticket, but the number of wakings if you win is so high you have credence 99% in winning) do you think it makes sense to talk about being "surprised" that you won, and if so would you, even with your 99% credence in winning the jackpot, still be surprised if you found you really won?

If you knew that you wouldn't lose your memory of the last awakening, would that change your answer? If you don't lose that last memory, then any mental anguish of falsely believing you won the lottery will stick with you, while (in the 1 in a million case that you won) all the many happy awakenings are forgotten except the last. This is one way in which a single awakening could be considered more important than the repeated awakenings of one person.

As for betting, maybe you can find the clear definition you're looking for, but so far I haven't seen it. At this point it seems we have agreed that bets will only be accepted or rejected on sunday, when everybody is a halfer.

Are you talking about new, unexpected bets after waking up?

 

[JeffJo]

Yes, the number of beauties matters, and it's because there is an experiment.

What if the number isn't known? Then neither can compute a confidence?

On sunday you are told the process that determines everything that follows. You only have a 1/1001 chance of winning, and a 1000/1001 chance of waking up next to the winner.

And the winner only had a 1/1001 chance of winning, and no chance of waking up next to the winner.

The question isn't asking you to assess these chances as they appeared to be before anybody was put to sleep. It is to assess them based on the current state of the experiment. I agree that this state does not include more information; it contains less. I argue that any change to the information state allows an update. The halfer argument, that there is nothing "new" = "more," is incomplete.

Drawing the conclusion that you are the winner with probability 1/2 after waking up is essentially oversampling,...

Only if you are under all of several mistaken impressions, that include (but are not limited to): (A) The question is about the experiment as a whole, and not the current state, and (B) It is asking for a statistical assessment where "sampling" is a meaningful concept, rather than probability where it is not.

The roadblock in this discussion, is that you are evaluating the confidence from outside the context in which it was asked; that is, in the context of the experiment as a whole. Beauty can only evaluate it on the inside, in the context of the current day.

...because you can never learn "I am asleep".

This problem was originated in the field of Philosophy, not Mathematics. I once heard a hyperbolic definition of Philosophy, as the science of forming a conclusion first, and finding reasons to justify it afterwords. With all due respect, that is what you are doing here.

How could it possibly matter how you might learn that a state, one you know isn't the current state, was occurring if it were?

Are you honestly saying that if we wake Beauty on (H,Tue), but take her to Disneyworld instead of interviewing her, that her confidence should be 1/3 when we do interview her on Monday or (T,Tue)? Simply because she would know it then?

What if the budget committee decides, on Sunday night, that they can't afford the trip and Beauty will be left asleep on (H,Tue). If we tell her this after she answers 1/3 in an interview, should she change to 1/2? Why? Does it matter if the meeting is scheduled for after the flip, so they only need to say decide if the coin is H?the question is moot?

I suggest to you, that the only important detail here is that Beauty can tell that she is in a specific state, and her confidence reflects just that state. How (of even if) she would she was in a different state is irrelevant.

Keep in mind, this argument only holds when you retain your memory of the sunday experiment.

And I suggest to you, that it on;ly applies to the experiment on Sunday, not on Monday.

"Repeating the experiment" requires ...

... invalid arguments that both sides use to their advantage. Being invalid doesn't mean you get the wrong answer, it just doesn't prove anything. Essentially, they are betting arguments. If they don't achieve resolution until Wednesday when Beauty is back outside of state where she was asked the question, they are invalid. If they evaluate your change in net worth relative to your net worth on Sunday, which is also outside, they are invalid. That's the "clear definition" you seek.

But I have proven the answer has to be 1/3, without invalid arguments.

 

[Stephen Tashi]

Suppose a person computes his "credence" about an event E by imagining a game where the ante is 1$ and if the event E happens person wins payoff $D$ (and keeps the ante). If the event E doesn't happen the person loses his ante. The person determines the value of payoff $D$ that he believes will make his expected gain zero. Then he solves the equation $(1-q) (-1) + q D = 0$ for $q$ and $q$ is define to the person's credence.

A problem with this definition is that different people can have different opinions about the value of $D$. If we specify that the person is "rational" then, in the case where the the probability of E can be computed by probability theory using given information,, we can assume a "rational" person computes $q$ from probability theory first and then uses that value to solve for $D$. That procedure makes the value of $q$ unique. However, if $q$ can be computed from the given information using probability theory, we could simply ask the person to compute $q$ using probability theory and forget about asking for "credence".

If we have a situation where the probability of E cannot be computed from the given information then must each "rational" person select the same value of $D$?

The mathematical solutions in this thread (as opposed to the philosophical ones) assume additional information not given in the problem and then apply probability theory to the information given in the problem plus those assumptions to compute $q$. One may obtain a solution in this manner, but where is the proof that such a solution is unique?

To guarantee that any "rational" Sleeping Beauty gets the same answer by mathematics, there must at least be some specific rules that specify what assumptions can be made. The "Principle of Indifference" ( i.e. that N mutually exclusive events for which we have no information to indicate one is more or less likely than another, each will be assigned "subjective" probability 1/N) is the only method that I detect in the various mathematical solutions. So we have the questions:

1) Is the principle of indifference specific enough that all "rational" people must agree whether it is applicable to a given set of events? (This seems to depend on interpreting the meaning of the word "information" - not the technical sense of Shannon information , but in the use of that word in common speech.)

2) Does applying the principle of indifference produce a unique answer to a given problem when a people are allowed free choice about which sets of "indistinguishable" events they will use in applying the Principle of Indifference?

(I don't claim to know the answers to those questions. People who claim to have "the" (unique) answer to the Sleeping Beauty Problem can comment on 2. )

The Sleeping Beauty problem is a problem of applying probability theory to an imagined real life situation and, as such, involves a certain amount of physics - at least it involves physical facts about the concepts of time that distinguish between "now" and "some other time".

For example, suppose we attempt to describe the states of the world that happen in the experiment with vectors of variables of the form:
(state of coin, state of SB, day of Week) then events that cause difficulty are:
A = (heads, awake,Monday)
B = (tails, awake Monday)
C = (tails, awake, Tuesday)

A straightforward interpretation of the experiment says that if the coin lands tails, both events B and C happen. So $P(B \cap C) = 1/2$.

But a "thirder" approach assumes (or concludes) that $\{A,B,C\}$ is a set of mutually exclusive events, each with probability 1/3.

To formulate an argument that reasons both about the events employed in the "thirder" approach and the events described in the experiment, we need a more refined vocabulary.

The apparent contradiction between the two uses of events B and C illustrates that the question posed in Sleeping Beauty problem does not ask about an event in the probability space of the experiment that is given in the problem. The question establishes the context "Sleeping Beauty is awakened..." with the implicit understanding that this event takes place on a particular day after a particular coin toss, to the exclusion of what happens on other days. The experiment gives a probability of Sleeping Beauty being awakened on various days without the implication that the awakenings B,C are mutually exclusive events.

The question posed by the Sleeping Beauty Problem can be asked using the notation for A,B,C for the events described in the experiment if we establish the context for the question by saying "One and only one of the events A,B,C is selected for consideration". However, this does not comment on what method is used to select the event. If some probability distribution were given for selecting the event, a unique answer to the question could be deduced from probability theory.

One "thirder" approach is to say that the subjective probability of selecting any given event from 3 events is 1/3 when we are given no information that indicates one of the events has a different probability of being selected than the others.

 

[JeffJo]

The mathematical solutions in this thread (as opposed to the philosophical ones) assume additional information not given in the problem and then apply probability theory to the information given in the problem plus those assumptions…

I have provided one such solution, so I can’t help but think you are referring to me; and failed to mention me specifically to avoid addressing my solution.

But I object to the characterization that assumptions were added. Yes, I varied the problem from the original by adding new participants, but (A) I have shown that the added participants are in functionally equivalent situations, and (B) each individual’s situation is functionally equivalent to the OP. So the additions are not assumed, they are proven equivalences.

The "Principle of Indifference" ( i.e. that N mutually exclusive events for which we have no information to indicate one is more or less likely than another, each will be assigned "subjective" probability 1/N) is the only method that I detect in the various mathematical solutions.

There are two forms of the PoI, sometimes called strong and weak. You described the weak, where there is nothing to indicate a difference, but also nothing to indicate a similarity. An example of the weak PoI is asking a child if (s)he likes coloring with crayons better than coloring with finger paint. The weak PoI says both have probability 50%.

In the strong version, you must demonstrate that the factors affecting the variations are identical. An example of the strong is my four Beauties, three of whom are awake. Each can arrive at the current state only by functionally equivalent processes: the day and the coin result do not both match what is written on her card.

The weak is flawed, because it is subjective where the strong is objective. As in...

2) Does applying the principle of indifference produce a unique answer to a given problem when a people are allowed free choice about which sets of "indistinguishable" events they will use in applying the Principle of Indifference?

(I don't claim to know the answers to those questions. People who claim to have "the" (unique) answer to the Sleeping Beauty Problem can comment on 2.)

And the difference in the application of the weak and strong versions to this question, is that this choice is only possible in the weak. So your question does not apply to the strong. If you can demonstrate indifference, the result of applying the PoI should be the same regardless of which set you apply it to. If you can only demonstrate that lack of a difference, the question is a good one.

The Sleeping Beauty problem is a problem of applying probability theory to an imagined real life situation and, as such, involves a certain amount of physics…

It is also one of applying probability theory. In that theory, a quantification such as “today is Monday” or “today is Tuesday” is random if you cannot determine which is correct, and both could be correct. There is no justification for a combination of the two in a strong PoI.

A straightforward interpretation of the experiment says that if the coin lands tails, both events B and C happen.

Sloppy. And part of the problem.

The neater interpretation, on Sunday, is that if the coin lands tails, both events WILL happen. Here I describe the events in a tense that is appropriate for the prior probability space. On Wednesday, we can say that both events DID happen. And that the posterior probability space appropriate for Wednesday has no “new information,” so it is identical to the prior.

But as you imply, an awake Beauty has to use the present tense. Whatever makes your B and C into a present-tense statement has to distinguish between B and C. What you overlook, is that the same is true after Heads. That is, the prior she needs to imagine requires four events:

A1 = (Heads, Monday will happen)
A2 = (Heads, Tuesday will happen)
B1 = (Tails, Monday will happen)
C1 = (Tails, Tuesday will happen)

Here, A1 and A2 represent the same future, as do B1 add C1. Inside the experiment, she needs to re-phrase these as "is happening," not "will happen." And the important part of however this is done, is that it separates these events into disjoint ones. Just how that is accomplished is the crux of the debate.

The halfer approach says that A2 is somehow eliminated from the event space. Without having to treat it, in Bayes Theorem, like you would any other eliminated event. But B1 and C1 morph into a quasi-same event: it is the same for determining the probability of H or T, but different for determining Monday or Tuesday. To me, this is absurd for two reasons: (1) The quasi-sameness is never explained; and (2) Event A2 still can happen. Somehow, halfers confuse "can't be observed by Beauty" with "can't happen." Which is also never explained.

 

[Stephen Tashi]

I have provided one such solution, so I can’t help but think you are referring to me;

I'm not. I'm referring to solutions like Dale's that are clear mathematical solutions based on specific assumptions.

 

[Dale]

That is why I have asked for a reference in the literature that explains that technical meaning.

Hi @PeterDonis,

I went through and found several sites discussing how credence is mapped to a wager. My favorite one was this one:
http://acritch.com/credence/
In this one the idea of credence of X is assessed on whether you would rather bet on X or on a biased roulette wheel with known bias. This actually is a fairly elegant approach because it doesn't depend on the payoff at all. It is a choice between two bets, would you rather bet on X or on a biased roulette wheel.

Section 3.3 here has a more standard formulation:
https://plato.stanford.edu/entries/probability-interpret/#SubPro
The "boils down" definition is the formulation that I have usually seen. I am generally a little hesitant/skeptical about the Stanford Encyclopedia of Philosophy entries, but this one seemed solid.

Michael Strevens has a detailed description of the process of assigning a wager to a credence in his chapter which is on page 301 of Arguing About Science (ed Alexander Bird, 2013). He also has some lecture notes entitled Notes on Bayesian Confirmation Theory which are more detailed but not as clear.

This paper was interesting:
http://download.springer.com/static/pdf/51/art%253A10.3758%252FBF03213487.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.3758%2FBF03213487&token2=exp=1497814716~acl=%2Fstatic%2Fpdf%2F51%2Fart%25253A10.3758%25252FBF03213487.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.3758%252FBF03213487*~hmac=f253b33bc93b2667ab101c91bf0934e5abddeee3f84376b56252d1b61fa67791
In addition to the "indifferent wager" technique mentioned in the first reference they also tested decision time as a method for assessing credence. One comment I found interesting in the introduction was a mention of some prior literature showing that statisticians were able to consistently articulate their credence regardless of the method whereas statistically naïve subjects were inconsistent in assessing their own subjective probability.

The Wikipedia entry is what I had usually taken as the standard
https://en.wikipedia.org/wiki/Credence_(statistics)
It defines the credence as the price at which a reasonable person would buy a $100 wager on X, but since what constitutes X is at dispute here I think that doesn't help. However, note that there is no complicated payout scheme involved, the wager is simple. None of these sources seem to support your approach.

 

[Dale]

The mathematical solutions in this thread (as opposed to the philosophical ones) assume additional information not given in the problem and then apply probability theory to the information given in the problem plus those assumptions to compute q. One may obtain a solution in this manner, but where is the proof that such a solution is unique?

Do you mean my assumptions that Beauty is rational and the experimenters are not malicious?