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

[Moes]

The previous sleeping beauty thread was closed asking me to provide a reference for the definition of credence that I would accept. I was not willing to accept a definition that says that credence must always follow the betting odds.https://www.anthropic-principle.com/preprints/beauty/synthesis.pdf

Here is a quote from page 19 in this article

“Now, we already know from other examples that when the number of bets depends on whether the proposition betted on is true, then the fair betting odds can diverge from the correct credence assignment”

I believe this is a good source that clearly states that the credence does not need to follow the betting odds.

I realized after the discussion that I was trying to argue using what's called anthropic reasoning (or indexical information). For anyone interested in anthropic reasoning I suggest a book called Anthropic Bias written by Nick Bostrom. The book is now free online here

https://www.anthropic-principle.com/q=book/table_of_contents/

 

[sysprog]

In the SB problem it is specified that SB is an epistemically sound reasoner. I think that any reasonable definition of that characteristic would have to include her reasoning at the outset that if she is awakened before the concluding Wednesday awakening, then there will be 3 possible heads-or-tails outcomes for the 2 coin tosses, with only 1 of them being heads, wherefore upon being asked what she thought was the probability of heads, she would say that it was 1/3.

If the examiner were to then say that the probability for a toss of a fair coin is 1/2, and ask her why she thinks that the probability for this one is 1/3, then she would say that it was because she's awake before the Wednesday concluding awakening, and that rules out the 2 heads outcome from the set of 4 possible outcome pairs, leaving 3, only 1 of which is heads, whence the 1/3 probability of heads.

If the examiner then said that he still thought that it was 1/2, she could remind him that by definition she was an epistemically sound reasoner, and leave it at that.

Or perhaps instead, she might ask him if he would like to bet on heads with her paying him 1.25 to 1 when heads came up, and him paying her 1 to 1 when tails came up, with the bettors using a Monte Carlo computer program to rapidly simulate as many iterations of the procedure as he could bankroll.

I imagine that the examiner would eventually re-think his position.

(@Peter Donis and @Dale made all of that abundantly clear in the other thread.)

 

[JeffJo]

The problem can be addressed without betting odds.

This is the Sleeping Beauty Problem, as first published by https://www.jstor.org/stable/3329167 ("Self-locating belief and the Sleeping Beauty problem," Adam Elga, Analysis, 60(2): 143-147, 2000.):

Some researchers are going to put you to sleep. During the two days 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?

A few observations:

• The introduction to the problem mentions "two days that your sleep will last," but never describes how it takes up those two days. I suppose one can assume that SB's sleep should last overnight, but that really isn't necessary for the problem.
• There is no order specified for the wakings. That is, there is no correspondence between the "first Tails waking" and the "only Heads waking."
• It mentions "when you are first awakened." Since there is no order specified, this wording is a little puzzling. It can't really mean the first of two (possible) awakenings, since the answer has to be the same for both. I take it to mean "before any information might be revealed." I think this because Elga's approach was to imagine what changes when information is revealed.

It was Elga who supposed the problem is the same as one where SB is always wakened on Monday, and on Tuesday if the coin landed on Tails. The problem, as discussed almost universally, includes those elements. And many of the disagreements seem to be about how those elements should be addressed. Betting odds is just one example one - the number of bets varies as mentioned by the cited article.

As I tried to discuss before, but was derailed by objections to an incidental and irrelevant, Elga's order and information are not necessary to solve the problem. So let me again pose a method to implement the actual problem Elga posed:

Some researchers are going to put you to sleep. Then they will flip two fair coins: a Dime and a Quarter.

They will perform this procedure:

1. If both coins are showing Heads, they end the procedure. If either is showing Tails, they continue.
2. They will briefly wake you up.
3. They will ask for your degree of belief that the Quarter is currently showing Heads.

1. After you provide an answer, they will put you to back to sleep with a drug that makes you forget this waking.

Regardless of where this procedure ended, they now turn the dime over and repeat it.

(I don't know why the editor wouldn't put all of tyhis is the same quote box. This should be step 4.

This is easy to solve. Regardless of whether the Dime has been turned over, you know that in Step 1 of orthe procedure that is being performed, there were four equally-likely combinations for (Dime, Quarter). They are (H,H), (T,H), (H,T) and (T,T). This is the "prior," and you do know that it existed even if you were not awake for it. But anytime you are awake in Step 3 of the procedure, you know that (H,H) has been eliminated. This is "new information," since there is a difference between the two states that you know about.

Your confidence in Heads should be 1/3.

 

[sysprog]

The problem can be addressed without betting odds

Yes, of course it can $-$ the betting scenario was not presented as necessary for solving the problem; it was an illustration, intended to make the correctness of the solution more obvious.

Your confidence in Heads should be 1/3.

I agree with this, but I think that your version of the procedure doesn't make it more perspicuous.

 

[JeffJo]

I agree with this, but I think that your version of the procedure doesn't make it more perspicuous.

If you agree with it, you shouldn't. You already see the point it makes. It is supposed to make "new information" clearer.

 

[sysprog]

If you agree with it, you shouldn't. You already see the point it makes. It is supposed to make "new information" clearer.

I agree with the conclusion; not with your re-versioning, which I think serves only to add obfuscatory rigamarole. No new information other than the fact that she has awakened on Monday or Tuesday is given to SB. She knows on Sunday that her answer when that happens will be 1/3.

 

[Moes]

This is "new information," since there is a difference between the two states that you know about.

Your confidence in Heads should be 1/3.

What exactly do you mean by “new information”? And how exactly does that information take you to the conclusion of 1/3?

The only way I could logically understand your argument is that what you mean to say is since when she wakes up she knows it can’t be both coins are facing heads (which you are calling new information), she should think that if the quarter landed heads it would be somewhat “surprising“ that she is up right now, since there was a chance that the dime would also be facing heads and she wouldn’t be woken up. Therefore she should assume the quarter probably landed tails and she was for sure going to be woken up.

I think this argument is wrong because I don’t think it should ever be surprising to be awake at the only time possible to be awake.

If I told you we were going to put you to sleep for a year and only wake you up one day sometime in the middle, when we wake you up you shouldn’t be surprised that it happens to be the one day that you are awake. You are obviously only able to think about the chances of you being awake on the one day that you are awake. If I told you the day you will be woken is July 1, you don’t gain “new information“ when you wake up that it’s not any other date. It was known all along that this was the one day you would wake up.

So I don’t consider her knowing that (H,H) has been eliminated “new information“ and even if you want to call it “new information” I don’t see how this information can be useful.The argument I’m making here is using what Nick Bostrom calls observation selection effects. If you don’t understand what I’m saying maybe try reading his book.

 

[JeffJo]

What exactly do you mean by “new information”? And how exactly does that information take you to the conclusion of 1/3?

I already said it. The prior is the state where there are four equally-likely combinations for what faces the coins are showing: (H,H), (T,H), (H,T) and (T,T). The new information is that one of those combinations has been eliminated, leaving (T,H), (H,T) and (T,T).

I think this argument is wrong because I don’t think it should ever be surprising to be awake at the only time possible to be awake.

Why do you think being surprised has anything to do with it?

Just before you were waken, you know that the two coins were, at that time, showing a combination. Now that you are awake, you know that the two coins are currently showing the same combination. That combination could be (T,T), which is just as likely as (H,T), and just as likely as (T,H). But it can't currently be (H,H). Three equally likely possibilities, for what the coins are showing at this moment in time, means that each has a 1/3 probability.

Honestly, you are making this more difficult than it is.

If I told you we were going to put you to sleep for a year and only wake you up one day sometime in the middle, when we wake you up you shouldn’t be surprised that it happens to be the one day that you are awake.

But if you had told me it could only happen on a weekday, I would know for a fact that it isn't Saturday. You seem to think my "degree of belief" should be the same for both Saturday and Thursday.

You are not placing yourself in the state of knowledge that corresponds to the procedure, you are using the state of knowledge of an outsider.

 

[Dale]

I believe this is a good source that clearly states that the credence does not need to follow the betting odds.

I am not too impressed with this source. First, the author is a philosopher writing a philosophy paper, not a statistician. So I find his credibility substantially weaker than the credibility of actual statisticians.

However, regardless of his credentials, I think that his argument is actually very weak, essentially a strawman.

First, they never actually defines credence, nor a way to measure it. So their claim that in certain circumstances acceptable betting odds do not reflect credence lacks a basis. Without a definition, who is to say that the circumstances lead to a difference between the credence and the rational betting odds?

Second, (and this is the strawman) he introduces the idea of a bet with a catch. This is a strawman because there are an infinite number of complicated wagers that could be offered for any proposition. Nobody in any of the other literature ever says that all possible wagers should represent the credence for the proposition. Indeed, such a scheme would be incoherent. When a specific wager is described in the literature it is always a very simple one, such as a straight bet on the spin of an unfair roulette wheel. The intent is to measure the belief in the proposition as directly as possible, not the belief in the proposition modified by a complicated betting scheme. So I reject the strawman argument that such a scheme is what other authors intend to convey with their linkage between credence and betting behavior.

Finally, and this is my personal opinion, in my opinion the odds that a person would accept for some wager with catches and conditions does indeed represent a credence. It is just the credence of a different proposition than simply "X is true".

 

[sysprog]

I already said it. The prior is the state where there are four equally-likely combinations for what faces the coins are showing: (H,H), (T,H), (H,T) and (T,T). The new information is that one of those combinations has been eliminated, leaving (T,H), (H,T) and (T,T).

Why do you think being surprised has anything to do with it?

Just before you were waken, you know that the two coins were, at that time, showing a combination. Now that you are awake, you know that the two coins are currently showing the same combination. That combination could be (T,T), which is just as likely as (H,T), and just as likely as (T,H). But it can't currently be (H,H). Three equally likely possibilities, for what the coins are showing at this moment in time, means that each has a 1/3 probability.

Honestly, you are making this more difficult than it is.

But if you had told me it could only happen on a weekday, I would know for a fact that it isn't Saturday. You seem to think my "degree of belief" should be the same for both Saturday and Thursday.

You are not placing yourself in the state of knowledge that corresponds to the procedure, you are using the state of knowledge'of an outsider.

I think that saying that SB's contigent awakening is "new information" to her introduces an inclarity. It's not new information with respect to her knowing on Sunday what her answer should or will be if/when she awakens on Monday or Tuesday. For that, she needs only the information that is given to her on Sunday.

 

[Moes]

I am not too impressed with this source. First, the author is a philosopher writing a philosophy paper, not a statistician. So I find his credibility substantially weaker than the credibility of actual statisticians.

I think there is an aspect to this problem that can't be solved just by using statistics, you need to also use logic. So I actually find the authors credibility stronger then statisticians who just deal with the numbers.

First, they never actually defines credence, nor a way to measure it. So their claim that in certain circumstances acceptable betting odds do not reflect credence lacks a basis. Without a definition, who is to say that the circumstances lead to a difference between the credence and the rational betting odds?

I never understood why credence should have to be defined using betting odds. I think the definition “level of belief“ is perfectly clear. Let’s say we forget about the word credence, is it not clear to you what a “level of belief“ would mean? The only thing that is not clear is how to get an exact measurement of your level of belief. Normally betting odds is just a good way to measure it. I don’t even think the other sources you brought argued with this.

Second, (and this is the strawman) he introduces the idea of a bet with a catch. This is a strawman because there are an infinite number of complicated wagers that could be offered for any proposition. Nobody in any of the other literature ever says that all possible wagers should represent the credence for the proposition. Indeed, such a scheme would be incoherent. When a specific wager is described in the literature it is always a very simple one, such as a straight bet on the spin of an unfair roulette wheel. The intent is to measure the belief in the proposition as directly as possible, not the belief in the proposition modified by a complicated betting scheme. So I reject the strawman argument that such a scheme is what other authors intend to convey with their linkage between credence and betting behavior.

I think what I said above might already explain why I would argue with this.

This is a strawman because there are an infinite number of complicated wagers that could be offered for any proposition

The intent is to measure the belief in the proposition as directly as possible

I think it is obvious that the point is to measure the belief in the way that most logically measures what she should actually believe, I don’t see why being as direct as possible would help.

Finally, and this is my personal opinion, in my opinion the odds that a person would accept for some wager with catches and conditions does indeed represent a credence. It is just the credence of a different proposition than simply "X is true".

I believe this means you simply have a different definition of credence. But what would you say her “level of belief” should be in the way that I am defining it?

 

[sysprog]

I think there is an aspect to this problem that can't be solved just by using statistics, you need to also use logic. So I actually find the authors credibility stronger then statisticians who just deal with the numbers.

Statisticians use logic & mathematics, and English (or another 'natural' language), just as other scientiists do.

I never understood why credence should have to be defined using betting odds.

It's not mandatory, and it's illustration; not definition.

I think the definition “level of belief“ is perfectly clear. Let’s say we forget about the word credence, is it not clear to you what a “level of belief“ would mean? The only thing that is not clear is how to get an exact measurement of your level of belief. Normally betting odds is just a good way to measure it. I don’t even think the other sources you brought argued with this.

The problem as stated doesn't go into any of this, or call for any of it. I think that it's the result of the efforts of some persons who held the untenable '1/2' position to steer the discussion into the realm of psychology.

I think what I said above might already explain why I would argue with this.

Hmm?

I think it is obvious that the point is to measure the belief in the way that most logically measures what she should actually believe, I don’t see why being as direct as possible would help.

The question is what is the epistemically sound thing to believe, and all that's required for answering that correctly is noticing that 1 of the 4 possibilities is outside of her possible before-Wednesday experience, leaving 3 possibilities, 1 heads, and 2 tails, for Monday or Tuesday, so that the correct answer on either of those days for the likelihood of heads is clearly 1/3.

I believe this means you simply have a different definition of credence. But what would you say her “level of belief” should be in the way that I am defining it?

I think that @Dale wasn't dwelling on any of that. The common definition is precise enough. SB, being an epistemically sound reasoner, could immediately when told the problem, respond that if she's asked on Monday or Tuesday what the likelihood of heads is, she'll believe that 1/3 is the correct answer.

 

[Moes]

I think that saying that SB's contigent awakening is "new information" to her introduces an inclarity. It's not new information with respect to her knowing on Sunday what her answer should or will be if/when she awakens on Monday or Tuesday. For that, she needs only the information that is given to her on Sunday.

I just want to point out that my argument is not just arguing that there is no “new information”. Even if you say that she had all the information from Sunday, I am arguing that even on Sunday, when she thinks about what her credence will be after the coin is tossed, thinking about the fact that she will be in an awakening should not be “information“ that can alter the probabilities. In a way the awakening will be nothing “new”. The fact that she will be awake is just a fact that must be true at the time that she will be able to think about the probabilities. I believe this is what halfers mean when they say there is no “new information“.I don’t think there is a point in arguing about whether the information is “new” in the way that you are arguing about it with JeffJo. PeterDonnis had a similar argument with him in a previous thread, but I’m not sure there is a real argument between them.

 

[Moes]

Just before you were waken, you know that the two coins were, at that time, showing a combination. Now that you are awake, you know that the two coins are currently showing the same combination. That combination could be (T,T), which is just as likely as (H,T), and just as likely as (T,H). But it can't currently be (H,H). Three equally likely possibilities, for what the coins are showing at this moment in time, means that each has a 1/3 probability.

The question is what is the epistemically sound thing to believe, and all that's required is noticing that 1 of the 4 possibilities will be outside of her experience before Wednesday, leaving 3 possibilities, 1 heads, and 2 tails, for Monday or Tuesday, so that the correct answer for the likelihood of heads is clearly 1/3.

Why do you consider it equally likely that she would be awake in each of the 3 awakenings? Since there was a 50% chance the coin would land heads when she is awake there should be a 50% chance that the coin landed heads and then the fact that it would be Monday and not Tuesday is just automatically true. Being awake is nothing interesting she new she would wake up sometime.

In other words the fact that she would wake up is not double as likely if the coin landed tails. It would just happen twice.

 

[sysprog]

I just want to point out that my argument is not just arguing that there is no “new information”. Even if you say that she had all the information from Sunday, I am arguing that even on Sunday, when she thinks about what her credence will be after the coin is tossed, thinking about the fact that she will be in an awakening should not be “information“ that can alter the probabilities. In a way the awakening will be nothing “new”. The fact that she will be awake is just a fact that must be true at the time that she will be able to think about the probabilities.

I agree with this, as presumably would others who say that 1/3 is the correct answer.

I believe this is what halfers mean when they say there is no “new information“.

The 'halfers' who bring up the absence of new information are apparently embracing the untenable position that new information beyond that of SB being awakened before Wednesday is necessary for correct analysis of the problem.

I don’t think there is a point in arguing about whether the information is “new” in the way that you are arguing about it with JeffJo.

PeterDonnis had a similar argument with him in a previous thread, but I’m not sure there is a real argument between them.

I think that there's a point in disagreeing with the notion that a more complicated reasoning path is necessary, when it seems clear that a simpler one will do just as well.

 

[Dale]

I think there is an aspect to this problem that can't be solved just by using statistics, you need to also use logic

I am also unimpressed with his logic. Logic needs to start with clear definitions and axioms. This author never defines credence.

I think the definition “level of belief“ is perfectly clear.

How do you assign a number to a level of belief?

Normally betting odds is just a good way to measure it. I don’t even think the other sources you brought argued with this.

Nor do I disagree.

But what would you say her “level of belief” should be in the way that I am defining it?

How are you defining it? I went through that whole paper and didn’t actually see a definition of credence

 

[sysprog]

Why do you consider it equally likely that she would be awake in each of the 3 awakenings?

Each outcome pair has an equal 1/4 chance to begin with. Elminating the 4th outhcome pair doesn't alter the equality of likelihood of the remaining 3 outcome pairs.

On Sunday she will know that if she awakens on Monday or Tuesday, the correct answer on being asked how likely 'the toss' (she won't know whether it's Monday's toss or Tuesday's toss) is to be heads, will be 1/3.

Since there was a 50% chance the coin would land heads when she is awake there should be a 50% chance that the coin landed heads and then the fact that it would be Monday and not Tuesday is just automatically true.

If she awakens before Wednesday, she can't discern whether it's Monday or Tuesday, but can tell that it's not Wednesday.

Being awake is nothing interesting she new she would wake up sometime.

She didn't know that her first awakening wouldn't be on Wednesday. If Wednesday is her first awakening after Sunday, then the 'both heads' outcome that's ruled out if she awakens on Monday or Tuesday has obtained.

In other words the fact that she would wake up is not double as likely if the coin landed tails. It would just happen twice.

There are 4 possible outcomes for the two possible tosses: th, tt, ht, hh, corresponding to awakening M only, both M and T, T only, or not until W. If she awakens, it's one of the first 3, and only one of those can be heads.

 

[Moes]

I am also unimpressed with his logic. Logic needs to start with clear definitions and axioms. This author never defines credence.

How do you assign a number to a level of belief?

Nor do I disagree.

How are you defining it? I went through that whole paper and didn’t actually see a definition of credence

I’m not sure I’m understanding you. You seem to agree that “level of belief” is a clear enough definition for credence, you just want to know how credence would be measured. To this I would say that her “level of belief” in something should be the same as whatever the probability of that thing happening (or having happened already) is. So in this case if when she is awake the probability of the coin having landed heads is 50% so her credence should be 1/2.

Since you are asking me how I would assign a number to a level of belief, I assume that when you ask how do I define credence you are asking how would I measure credence.

I think the way I’m saying to measure credence was obvious to the author of the paper and so there was no need to define it. The definition is simply a level of belief. To measure it you simply need to figure out what the probability is.

 

[Moes]

She didn't know that her first awakening wouldn't be on Wednesday. If Wednesday is her first awakening after Sunday, then the 'both heads' outcome that's ruled out if she awakens on Monday or Tuesday has obtained.
There are 4 possible outcomes for the two possible tosses: th, tt, ht, hh, corresponding to awakening M only, both M and T, T only, or not until W. If she awakens, it's one of the first 3, and only one of those can be heads.

I am not understanding this. I think you are dealing with a different problem then I am. In my version of the sleeping beauty problem there is no chance she wouldn’t be woken up until Wednesday.

 

[sysprog]

How do you assign a number to a level of belief?

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.

 

[Dale]

You seem to agree that “level of belief” is a clear enough definition for credence

I do not agree with that. “Level of belief” is a description of what a credence is, not a definition. Using that statement I cannot apply the definition to determine if a specific number is indeed the credence for some proposition. In fact, there is nothing in that description to even indicate that a credence is numerical. A level of belief could be some sort of ordinal measure.

I would say that her “level of belief” in something should be the same as whatever the probability of that thing happening (or having happened already) is.

That isn’t a credence, that is just a probability.

Since you are asking me how I would assign a number to a level of belief, I assume that when you ask how do I define credence you are asking how would I measure credence.

I am asking both.

I think the way I’m saying to measure credence was obvious to the author of the paper and so there was no need to define it.

Seems like a big omission to me, especially for a philosopher. To me, this source is not terribly impressive. The only reason that I think we are discussing it is because it says what you want to hear, not that it is in itself a particularly good reference. It doesn’t even explicitly define credence and its argument against the betting approach is a strawman.

 

[Moes]

I do not agree with that. “Level of belief” is a description of what a credence is, not a definition. Using that statement I cannot apply the definition to determine if a specific number is indeed the credence for some proposition. In fact, there is nothing in that description to even indicate that a credence is numerical. A level of belief could be some sort of ordinal measure.

That isn’t a credence, that is just a probability.

I am asking both.

Seems like a big omission to me, especially for a philosopher. To me, this source is not terribly impressive. The only reason that I think we are discussing it is because it says what you want to hear, not that it is in itself a particularly good reference. It doesn’t even explicitly define credence and its argument against the betting approach is a strawman.

So when she is awake what would you say the “probability” of the coin having landed heads is?

 

[Dale]

So when she is awake what would you say the “probability” of the coin having landed heads is?

Are you asking for $P(heads)$ or $P(heads|awake)$? Your question is unclear.

 

[Moes]

In the SB problem it is specified that SB is an epistemically sound reasoner. I think that any reasonable definition of that characteristic would have to include her reasoning at the outset that if she is awakened before the concluding Wednesday awakening, then there will be 3 possible heads-or-tails outcomes for the 2 coin tosses, with only 1 of them being heads, wherefore upon being asked what she thought was the probability of heads, she would say that it was 1/3.

If the examiner were to then say that the probability for a toss of a fair coin is 1/2, and ask her why she thinks that the probability for this one is 1/3, then she would say that it was because she's awake before the Wednesday concluding awakening, and that rules out the 2 heads outcome from the set of 4 possible outcome pairs, leaving 3, only 1 of which is heads, whence the 1/3 probability of heads.

If the examiner then said that he still thought that it was 1/2, she could remind him that by definition she was an epistemically sound reasoner, and leave it at that.

Or perhaps instead, she might ask him if he would like to bet on heads with her paying him 1.25 to 1 when heads came up, and him paying her 1 to 1 when tails came up, with the bettors using a Monte Carlo computer program to rapidly simulate as many iterations of the procedure as he could bankroll.

I imagine that the examiner would eventually re-think his position.

(@Peter Donis and @Dale made all of that abundantly clear in the other thread.)

Sorry I should have realized we were dealing with different problems already from this. In my problem there is only one coin toss. From how I understand your version I agree the obvious answer is 1/3. You obviously didn’t follow the previous threads.

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

 

[sysprog]

I am not understanding this. I think you are dealing with a different problem then I am. In my version of the sleeping beauty problem there is no chance she wouldn’t be woken up until Wednesday.

In the foregoing, I have been incorrectly, albeit to no consequence as to conclusion, using the two days as if the problem had been stated as having two coin tosses, one on Monday and one on Tuesday $-$ mea culpa sed nihil interest $-$ sorry if that was confusing or misleading to some readers/contributors. I concede that I think that it does make the untenability of the '1/2' position a bit less obvious.

 

[Moes]

Are you asking for $P(heads)$ or $P(heads|awake)$? Your question is unclear.

This is what you need to decide. That is exactly the question. I think the question is clear. The question is when sleeping beauty is awake should she assign 50% probability to heads or should she assign a probability of 1/3?

Meaning should she calculate the probability based on the probability of a fair coin or based on how many times she will be woken up? What should decide her “level of belief“ what should she actually believe? That the coin probably landed tails or heads and tails were equally probable?

 

[sysprog]

This is what you need to decide. That is exactly the question. I think the question is clear. The question is when sleeping beauty is awake should she assign 50% probability to heads or should she assign a probability of 1/3?

$P(heads)$ = 1/2, and $P(heads|awake)$ = 1/3, because she has 2 awakenings if tails, and only 1 if heads, making the chance that this awakening is a tails awakening 2 out of 3, and that it is a heads awakening only 1 out of 3.

 

[Dale]

This is what you need to decide. That is exactly the question. I think the question is clear. The question is when sleeping beauty is awake should she assign 50% probability to heads or should she assign a probability of 1/3?

Meaning should she calculate the probability based on the probability of a fair coin or based on how many times she will be woken up? What should decide her “level of belief“ what should she actually believe? That the coin probably landed tails or heads and tails were equally probable?

Then the relevant probability is $P(heads|awake)=1/3$. There is no doubt whatsoever about that probability.

 

[Dale]

$P(heads)$ = 1/2, and $P(heads|awake)$ = 1/3

There is no doubt about either of these probabilities.

 

[Dale]

To all in this thread. There have already been several very long threads on the Sleeping Beauty problem. Those previous threads have been closed, and if this is to simply be another then it will be closed quickly.

If this thread is to remain open then it must be different from these others. The OP is about a specific reference and its concept of credence. That is a more focused and a distinct topic which can remain.

 

[Moes]

There is no doubt about either of these probabilities.

Then the doubt is which one of them to use when she is awake and reasoning about the probabilities. But I actually don’t think it’s a doubt she cannot be any more sure that the coin landed tails then that it landed heads. So her credence is 1/2.

 

[Dale]

Then the doubt is which one of them to use when she is awake and reasoning about the probabilities. But I actually don’t think it’s a doubt she cannot be any more sure that the coin landed tails then that it landed heads. So her credence is 1/2.

So, what in the definition of credence requires the use of the unconditional probability? Surely people are allowed to let their degree of belief depend on the conditions.

When she is awakened she clearly knows that she is awake, so why should she be forced to believe $P(heads)$ and not $P(heads|awake)$?

 

[Moes]

To all in this thread. There have already been several very long threads on the Sleeping Beauty problem. Those previous threads have been closed, and if this is to simply be another then it will be closed quickly.

If this thread is to remain open then it must be different from these others. The OP is about a specific reference and its concept of credence. That is a more focused and a distinct topic which can remain.

I don’t think I should keep arguing anymore but even if I do I would like if you can just tell me to stop when you don’t think we are getting anywhere. I would like this thread to remain open so others can later give there opinions.

 

[Dale]

I don’t think I should keep arguing anymore but even if I do I would like if you can just tell me to stop when you don’t think we are not getting anywhere. I would like this thread to remain open so others can later give there opinions.

Then let’s discuss credence and not sleeping beauty.

 

[Moes]

Then let’s discuss credence and not sleeping beauty.

Ok, one way I think I could explain it, is that you can only let your belief depend on the conditions if the conditions could have been different. In this case she could not have been thinking about the probabilities when she was sleeping. So the condition that she is awake can not be used to decide her credence. The probability of the coin toss was 50/50 the condition that she is now awake shouldn’t change anything. So the probability should remain 50/50