B What should Sleeping Beauty's credence be in a life-or-death coin toss?

[Moes]

Once a week someone flips a coin, and if it's heads they turn a light on and leave it on for one day, then they go back and turn it off, and if it's tails they turn a light on for two days they go back and turn it off. You're aware of this, but you don't remember which day of the week they flip the coin on. You walk into the room one day and see the light is on. What is the probability the coin flip was tails?

Yes, this is a simple example where the probability is obviously 2/3. This was never questionable. This would be like a random person walking into a room where he knows the sleeping beauty experiment is taking place, but doesn’t have any other information. If he sees sleeping beauty awake he should think the probability that the coin landed tails is 2/3.

Maybe it would help if we drop the weird thing about being awake and having no memory

The loss of memory is the key point in this problem.

This is exactly what I think the problem is. People think the sleeping beauty problem is just another mathematical probability question. They don’t realize that the loss of memory adds philosophical type of questions to the problem. I guess mathematicians are just not the right type of people to ask about these questions. But I’m surprised how anyone can really believe the answer is not 1/2. Just thinking about myself in sleeping beauty’s situation the answer seems obvious to me.

 

[sysprog]

I guess mathematicians are just not the right type of people to ask about these questions.

You don't have to be a mathematician to solve this, but it doesn't hurt.

 

[sysprog]

I’m surprised how anyone can really believe the answer is not 1/2.

How can you really believe that it's not 1/3, when in this thread you have shown repeatedly that you know that 1/3 is the correct answer?

Just thinking about myself in sleeping beauty’s situation the answer seems obvious to me.

If you pretend that SB is asked what's the probability for a random coin toss, then the 1/2 answer is obvious, but that's not the postulated situation, and you've shown that you know that, and that you know that the perhaps less obvious correct answer in the scenario as described is 1/3.

 

[Dale]

But I’m surprised how anyone can really believe the answer is not 1/2.

That sounds like something you should work on. You may not agree with an answer or an argument, but you should be able to get out of your own head and understand other people enough to see how they can believe something you don’t.

In this case the argument is exceptionally simple: I believe that the credence of a person is measured by the bet they would take as described in the blog I linked to, 1/3 is the break-even for that specific bet, and since she is rational and rational people don’t want to lose money that is the bet she would take, and hence her credence.

If you cannot understand my belief then you aren’t even trying to do so. You even agree on everything except the definition of credence. But when someone clearly explains what they mean by a word and then use that word exactly as they have explained then any lack of understanding is on the other part.

Again, you don’t have to agree with the thirder position, but at this point if you fail to understand it that is on you.

 

[hutchphd]

This is exactly what I think the problem is. People think the sleeping beauty problem is just another mathematical probability question.

But at the point it is not " just another mathematical probability question" it is no longer subject to rigorous analysis and you can torture it to produce any desired answer.
How many angels can dance on the head of a pin?

 

[Dale]

But at the point it is not " just another mathematical probability question" it is no longer subject to rigorous analysis and you can torture it to produce any desired answer.
How many angels can dance on the head of a pin?

And since the answer in the end is a number then math must enter in at some point.

 

[JeffJo]

But I’m surprised how anyone can really believe the answer is not 1/2. Just thinking about myself in sleeping beauty’s situation the answer seems obvious to me.

This opinion is based on the misconception that SB "receives no (new) information" by being wakened. Yet I have seen no definition of what halfers who say this think "new information" means, even from the ones who keep demanding that I supply one.

They have said that because she knew she would be wakened all along, nothing she "learns" from being awake tells her anything she didn't know on Sunday Night. But they haven't defined how this distinction means anything to probability theory. And it is easy to remedy: In Elga's reformulation (it isn't the original problem, which is another thing nobody wants to recognize), wake her both days. On Monday, and on Tuesday if the result is Tails, interview her with the question about confidence. But on Tuesday, if the result is Heads, take her to Disneyworld without asking her anything about the coin.

This way, if she is interviewed, she knows that there are four possible states under which she could be awake. She also knows that each had the same probability of being the current state when she woke up, and that one is now ruled out by the fact that she is not at Disneyworld.

So now her "awake" knowledge includes something that was not certain when she went to sleep on Sunday. The answer, when she is interviewed, has to be 1/3.

The point here is that "new information" includes anything she can learn about the current state, including what she knows it isn't. It does not matter what would happen in the state she knows does not correspond to the current state, just that she knows it isn't the current state. Being awake and interviewed supplies this information.

+++++

Edit: And to make it more intuitive, use the common trick that halfers think makes their answer more intuitive. Wake her every day for a year. On Jan 1, and on every day after that if the coin result was Tails, ask her for her credence about the coin. But on those other 364 days, if the coin result is Heads, take her on a different outing and don't ask her about the coin.

 

[Dale]

Yet I have seen no definition of what halfers who say this think "new information" means,

I have also not seen a definition about “credence” that leads to the halfer position. I have seen the definition that credence is a degree of belief, but that definition doesn’t lead to any number, let alone specifically to 1/2. And the reference in the OP didn’t provide a definition, it just objected to using a betting scheme in either the definition or measurement of credence.

While I understand the halfer argument, I think it is not well founded. With the key issue being an imprecise definition of credence and a second issue being not using the conditional probability.

 

[JeffJo]

I have also not seen a definition about “credence” that leads to the halfer position. I have seen the definition that credence is a degree of belief, but that definition doesn’t lead to any number, let alone specifically to 1/2. And the reference in the OP didn’t provide a definition, it just objected to using a betting scheme in either the definition or measurement of credence.

While I understand the halfer argument, I think it is not well founded. With the key issue being an imprecise definition of credence and a second issue being not using the conditional probability.

It is my opinion that halfers want "credence" to have a different meaning than "probability," so that they can dismiss arguments that use probability theory.

 

[Dale]

It is my opinion that halfers want "credence" to have a different meaning than "probability," so that they can dismiss arguments that use probability theory.

As long as they are clear, that is fine. Such definitions can indeed make 1/3 not the answer, but it becomes difficult to get 1/2 or any other number also. With such a definition then how do you get any number? Then if it isn’t a probability then how do you get a number between 0 and 1? And then how do you specifically get 1/2? So often I think that such definitions don’t, in fact, support the 1/2 argument. Specifically the "level of belief" definition, by itself, is insufficient to get any number, including 1/2, IMO.

 

[Moes]

That sounds like something you should work on. You may not agree with an answer or an argument, but you should be able to get out of your own head and understand other people enough to see how they can believe something you don’t.

Yes, this is exactly what I’m trying to do, but I’m not going to make believe I understand something when I don’t.

In this case the argument is exceptionally simple: I believe that the credence of a person is measured by the bet they would take as described in the blog I linked to, 1/3 is the break-even for that specific bet, and since she is rational and rational people don’t want to lose money that is the bet she would take, and hence her credence.

If you cannot understand my belief then you aren’t even trying to do so. You even agree on everything except the definition of credence. But when someone clearly explains what they mean by a word and then use that word exactly as they have explained then any lack of understanding is on the other part.

Again, you don’t have to agree with the thirder position, but at this point if you fail to understand it that is on you.

I think I actually do understand your argument about credence, but simply disagree with it. Others in this thread seem to have a different argument for why they believe the answer is 1/3. That is what I don’t understand.

I have seen the definition that credence is a degree of belief, but that definition doesn’t lead to any number, let alone specifically to 1/2

For example: for 2-valued, 0 = no and 1 = yes; for 3-valued, 0.5 = undecided; for 5-valued, 0.25= probably not, 0.75 = probably.

I believe this way of leading to a number is good enough. All I’m trying to say is that she should be 0.5 = undecided . It seems like you or at least others in the thread believe she should think the probability of heads is “probably not”

It is my opinion that halfers want "credence" to have a different meaning than "probability," so that they can dismiss arguments that use probability theory.

No, the problem here is how to apply probability theory.

How can you really believe that it's not 1/3

The point here is that "new information" includes anything she can learn about the current state, including what she knows it isn't. It does not matter what would happen in the state she knows does not correspond to the current state, just that she knows it isn't the current state. Being awake and interviewed supplies this information.

I will try once more to explain the simple logical reasoning that leads to an answer of 1/2.

The information that sleeping beauty has before she is put to sleep is that she will soon be awakened guaranteed. She is put to sleep and sure enough soon awakened. She is told there was a coin toss but it had no effect on these events. Whether the coin landed heads or tails she was to be put to sleep and soon awakened and this is all that she knows happened.

How could she be anything other than completely unsure how the coin landed?

The second awakening does nothing. The loss of memory basically just takes her back in time to replay an event for a second time.

When thinking about it in this way the logic here just seems way too obvious to me to argue about.

 

[Dale]

I think I actually do understand your argument about credence, but simply disagree with it.

That is fine. Then I would recommend saying that you disagree with it, rather than you don’t understand it.

I believe this way of leading to a number is good enough. All I’m trying to say is that she should be 0.5 = undecided

I agree that is a perfectly fine way of leading to a number, but it does not follow in any way from the “level of belief” definition. I could just as well use a scale where “undecided” is just a word, or where it is assigned a numerical value of 5 or 50 or even $\pi$. All of those scales would also represent levels of belief, and in fact the first two are much more typical representations of latent ordinal scales than a 0 to 1 scale.

this is all that she knows happened.

This is not correct. She also knows that she will be awakened twice as often on tails than on heads. How you choose to deal with that knowledge clearly differs from me, but that is something that she does know.

The second awakening does nothing.

This is also not correct. The second awakening doubles the frequency of awakening for tails. How you choose to deal with that fact clearly differs from me, but it isn’t “nothing”.

When thinking about it in this way the logic here just seems way too obvious to me to argue about

I understand your argument, but it is not very compelling. It neglects some key features of the scenario and it results in a losing betting strategy for the usual credence bet.

 

[JeffJo]

The information that sleeping beauty has before she is put to sleep is that she will soon be awakened guaranteed. She is put to sleep and sure enough soon awakened. She is told there was a coin toss but it had no effect on these events. Whether the coin landed heads or tails she was to be put to sleep and soon awakened and this is all that she knows happened.

How could she be anything other than completely unsure how the coin landed?

The second awakening does nothing. The loss of memory basically just takes her back in time to replay an event for a second time.

When thinking about it in this way the logic here just seems way too obvious to me to argue about.

You are confusing knowledge about how the random state is created and changed, with knowledge about what that state is at a moment in time.

We do understand that she had full knowledge on Sunday Night about the eventuality of being awake. That is not the knowledge that allows one to update the probability space that applies to a state. "New information" refers to the change in that probability space.

That's why I proposed a different way to implement the original problem using two coins, and an alternative to Elga's implementation of it where she is always wakened on both days.

It is the possible portion of the experiment, where she would be awake if the result is Tails but not if it is Heads that "does something." This moment exists, whether or not SB is awake during the experience.

 

[JeffJo]

Again, in the two-coin version, there is one probability space (I'll call it PS1) that applies to when the decision is made to wake you. (Note: I am deliberately using "you" instead of "SB" to distinguish my version.) The sample space contains the four outcomes {HH, HT, TH, TT}. The probability for each is 1/4.

But when you are awake, you know that the probability space has changed (to PS2). The sample space is now {HT, TH, TT}, and the probabilities are updated to (1/4)/[(1/4)+(1/4)+(1/4)]=1/3. The "new information" has nothing to do with knowing that you will reach this state, it is that PS2 applies to the current state.

 

[Office_Shredder]

Moe, since all the information is available on Saturday, she can answer the following question on Saturday.
What is your credence on each day we would wake you up?
What is your credence on each day we let you sleep?If your answers are 1/2 and 1, you need a really good justification.

If not being woken up increases your credence it's heads, being woken up must decrease it.

 

[sysprog]

I think I actually do understand your argument about credence, but simply disagree with it. Others in this thread seem to have a different argument for why they believe the answer is 1/3.

I think that the reasoning expressed by @Dale is functionally the same as that of others who expressed the same conclusion.

No, the problem here is how to apply probability theory.

In that regard, it's a very simple arithmetic problem: 1 chance out of 3 = 1/3 probability.

The information that sleeping beauty has before she is put to sleep is that she will soon be awakened guaranteed. She is put to sleep and sure enough soon awakened. She is told there was a coin toss but it had no effect on these events.

The coin toss has the effect described in the problem statement: if heads, 1 awakening; if tails, 2 awakenings.

Whether the coin landed heads or tails she was to be put to sleep and soon awakened and this is all that she knows happened.

She also knows that she is to be awakened 1 time if heads, and 2 times if tails.

How could she be anything other than completely unsure how the coin landed?

There is 1 day's chance for heads, and 2 days' chances for tails.

The second awakening does nothing. The loss of memory basically just takes her back in time to replay an event for a second time.

Without the loss of memory, on a second awakening she would know that it wasn't the first, and therefore that the toss consequently was tails with probability of 100%.

When thinking about it in this way the logic here just seems way too obvious to me to argue about.

I think that some people allow themselves to be misled by the glaring 50%-probability nature of a coin toss, into failing to give due regard to the fact that SB knows that there is 1 day's chance for heads and 2 days' chance for tails.

 

[hutchphd]

@sysprog, Drop the Mic.

 

[Dale]

@Moes here is one additional thing to consider. In science we like measurable quantities because we can use the scientific method on them. The odds at which someone accepts a bet is measurable, and therefore of scientific interest. With the assumption that people’s betting behavior on an uncertain proposition relates to their level of belief about the proposition, then it becomes a measurement of their level of belief.

Now, you can use a different definition of credence, but the fact remains that the betting odds are a measurable quantity of interest related to a person’s level of belief.

We all agree that the break-even betting odds is 1/3. So even if you disagree with calling it her credence, the simple fact remains that 1/3 is a scientifically interesting quantity related to her belief.

 

[JeffJo]

Dale, I agree 100% with your conclusions. But not with your reasons.

The issue with the betting argument, is whether the entire procedure involves exactly one bet, or one-or-two bets, depending on the coin. Again, I agree with the result that one-or-two bets is correct, but you can't divorce the "what is your confidence" question from the "how many bets" one. So demonstrating that 1/3 agrees with the one-or-two bets option, while ignoring that 1/2 agrees with the exactly one bet option, does not prove anything.

And that's why I tried to re-cast the solution for the original problem. Elga's "Always-Monday, Tuesday-on-Tails" solution seems to be equivalent to the original problem, but introduces the number of bets question. Using my two coins circumvents that issue.

 

[Dale]

you can't divorce the "what is your confidence" question from the "how many bets" one.

I am not sure why you think I am doing that.

But my previous post was not an argument with a conclusion. I was just pointing out the fact that measurable quantities are of scientific interest. The point isn’t a conclusion but simply pointing out that redefining the word “credence” to not be 1/3 in this case doesn’t reduce the scientific importance of the quantity that is 1/3 in this case

 

[JeffJo]

I am not sure why you think I am doing that.

But my previous post was not an argument with a conclusion.

We all agree that the break-even betting odds is 1/3. So even if you disagree with calling it her credence, the simple fact remains that 1/3 is a scientifically interesting quantity related to her belief.

If, over the entire procedure, you risk losing only $1 on either result, and win only $1 on if you are right on either result, the break-even odds are 1:1 (the probabilities are 1/2).

If you risk losing only $1 on heads, but $2 on tails? And win only $1 on if you are right on heads, but $2 on Tails? Then the odds are 2:1 and 1:2 (or probabilities are 1/3 and 2/3). You concluded that "we all agree that the break-even [probability] is 1/3. That is actually an issue that is contended by some.

 

[Dale]

If, over the entire procedure, you risk losing only $1 on either result, and win only $1 on if you are right on either result, the break-even odds are 1:1 (the probabilities are 1/2).

If you risk losing only $1 on heads, but $2 on tails? And win only $1 on if you are right on heads, but $2 on Tails? Then the odds are 2:1 and 1:2 (or probabilities are 1/3 and 2/3).

Yes, that is all clear and has been discussed previously. It is understood and I felt didn’t need to be re-hashed every time the number 1/3 is used.

You concluded that "we all agree that the break-even [probability] is 1/3. That is actually an issue that is contended by some.

”We all” is referring to those here. Specifically @Moes does not contend that issue and has agreed with it several times. Again, that wasn’t a conclusion, just a reference to the opinions already previously expressed in the conversations here.

 

[JeffJo]

And again, I agree with your conclusion that confidence should be 1/3. But I disagree with using any argument based on betting, since there are those who will disagree with how the betting should proceed. Whether or not they post here, their reasoning appears to be similar to arguments that are used here for what "confidence" should mean.

BUT THE ISSUE CAN BE AVOIDED.

You are put to sleep, and then a dime and a quarter are flipped. If either is showing tails, you are (1) wakened, (2) asked for your probability/confidence/whatever that the quarter is currently showing Heads, (3) amnesia'ed, and (4) put back to sleep. Then the dime is turned over. If either coin is now showing Tails, the same four steps are performed.

This exactly matches the specifications in original problem. So does Elga's version with the steps mandatory on Monday, and based on the single coin on Tuesday. But this introduces issues about betting, and what "confidence" means, that are not present in my version.

 

[Dale]

But I disagree with using any argument based on betting

And again, I wasn’t making an argument or a conclusion. I was merely pointing out the fact that the odds someone will accept is a measurable quantity and hence is scientifically interesting, even if it isn’t “credence”. So redefining credence to something else doesn’t change the interest in the quantity that is 1/3.

Similarly, you are free to pursue non-betting arguments, but that also does not change the fact that the odds someone is willing to accept is measurable and thus of scientific interest. That doesn’t impact your argument whatsoever.

You appear to have missed my intention. I apologize for the lack of clarity. I am a scientist at heart so measurable quantities are of primary interest. I am simply pointing that out.

This exactly matches the specifications in original problem.

Have you ever had a halfer agree to that statement?

 

[JeffJo]

And again, I wasn’t making an argument or a conclusion.

And again, "I/You/We/They agree that the break-even betting odds is 1/3" is a conclusion. About both the betting odds, and who agrees.

Have you ever had a halfer agree to that statement?

I've never had anybody agree or disagree, whether thirder, halfer, or on-the-fencer. I thought this forum might evoke more direct discussion than some others, but it seems to be to mired in minutia.

But I'll clarify the statement:

• Original Problem: Some researchers are going to put you to sleep. During the [irrelevant time period because it is not mentioned again] that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?
• Adam Elga's modification, for the purposes of solution: Change the subject from you to a volunteer we can call SB. Put SB to sleep on Sunday Night, then flip the coin. Always wake SB on Monday, and again on Tuesday if the coin result was Tails. Change "degree you believe" to "credence."
• My modification: Flip a second coin at the same time the first is flipped. Wake you if either coin is showing Tails. After putting you back to sleep, turn the second coin over and repeat the same process. Ask only about current state of first coin, noting that "the outcome is Heads" means the coin is currently showing Heads.

What needs to be agreed on, is whether either modification matches the intended problem. Mine did not change anything about it (the question asks about the same occurence, just at a different time), it just created an unbiased framework within which to execute it as worder. Elga's added a similar framework, and imposed an ordering, which may or may not be seen as a bias.

Then, one needs to solve it. Elga's solution is what gets mired in minutia, much of which centers on either the ordering or the word "credence." Mine has a clear solution in probability theory.

• Yes, it is clear. Part of the minutia is what "new information" means. It means a change in the what you know to be the probability space, not what either you or SB knows about the framework.

 

[hutchphd]

Mine has a clear solution in probability theory.

This (being comment #85) would seem self-referentially untrue.

 

[Dale]

And again, "I/You/We/They agree that the break-even betting odds is 1/3" is a conclusion. About both the betting odds, and who agrees

This is silly. Simply referencing a previous statement is not a conclusion. If this were the case then every reference in a scientific paper would be considered a conclusion. But it is irrelevant to the thread, so I won’t argue further about it. Repeat it again if you feel you must have more silliness.

 

[Office_Shredder]

I feel like this is really hitting the nail on the head. What can you do with the number 1/2 here in any context?

 

[JeffJo]

This (being comment #85) would seem self-referentially untrue.

Not sure what you mean. None of the previous 84 comments have even touched on what I said was "clearly true." It was about how a different formulation of the problem corresponds to the original. I keep trying to get discussion to happen, but the only comments I have gotten are about incidental issues that have nothing to do with it. Like...

This is silly. Simply referencing a previous statement is not a conclusion. If this were the case then every reference in a scientific paper would be considered a conclusion. But it is irrelevant to the thread, so I won’t argue further about it. Repeat it again if you feel you must have more silliness.

Yes, it is silly. You were referencing previous statements where you, and others, drew and mutually agreed with a conclusion. It is very silly to keep insisting that there was no conclusion, by repeating over and over that the referencing statement itself isn't a conclusion.

Again, yes, that is obvious. Just as obvious as the fact that it was talking about a conclusion. That's all I meant. That I agree with the same conclusion you all do, but not with some of the reasons I've seen used to justify it. Not that the conclusion originates in the statement "We all agree with XXX." It was made when you agreed to XXX.

 

[Stephen Tashi]

Maybe it would help if we drop the weird thing about being awake and having no memory.

Once a week someone flips a coin, and if it's heads they turn a light on and leave it on for one day, then they go back and turn it off, and if it's tails they turn a light on for two days they go back and turn it off. You're aware of this, but you don't remember which day of the week they flip the coin on. You walk into the room one day and see the light is on. What is the probability the coin flip was tails?

Can the person flip the coin on the last day of the week?

Are we assuming that not knowing which day of the week the coin was flipped is the problem posers way of stating that each day of the week has an equal probability of being the day the coin was flipped? If so, this is must be justified by a cultural convention - e.g. When we are solving a problem on an exam that is not worded with complete clarity, we know we must assume enough information to solve the problem.