The Actual Sleeping beauty Problem

[JeffJo]

TL;DR Summary

Most versions of the Sleeping Beauty Problem that you may have seen introduce elements Adam Elga added to it, to make his solution eaasier. There is a better way.

First, a little history. I'm assuming you are familiar with how the details are governed, so I'll only list what the details were:

1. The problem originated in an unpublished (until much later) work by Arnold Zuboff.
   1. The experiment was to last a trillion days.
   2. Depending on a fair coin toss (Heads or Tails was not specified), the subject (who I will call SB) would be wakened either on each of the trillion days, or on a single, randomly-chosen day in that period.
   3. The question was "How should [SB] answer the question of how many times [she is] being awakened?
2. In 2000, Adam Elga recognized the hyperbole in the trillion-day version, and pared it down considerably.
   1. He proposed either one, or two, awakenings.
   2. He was ambiguous about the time period. He said the experiment was to occur over two days, but no awakenings were assigned to a specific day in that period.
   3. The one-awakening procedure would occur after a Heads result, and the two-awakening procedure after a Tails result.
   4. The question was "To what degree ought [SB to] believe that the outcome of the coin toss is Heads?"
3. But in order to delineate the events in his solution, Elga:
   1. Re-introduced one potential awakening per day.
   2. Removed Zuboff's random-day specification for the single awakening, fixing it on the first day.
   3. Added a word that becomes important only in his solution.

Just to make it clear that I am not changing anything, here is the problem as stated by Elga:

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?

Most of the arguments - including one recently closed in this forum - focus on points that, in my opinion, are irrelevant.

• There are various translations of what Elga's "degree of belief" should mean. To be fair, Elga does use the term "credence" for it in the solution, but he also treats it as a straight-forward probability.
• He added the order "Always awaken on Monday, but only on Tuesday after Tails."
• In his solution, it becomes clear that "when first awakened" means "before anybody might tell SB that it is Monday, or that the coin landed on Heads." He uses such revelations to reduce the probability problem to ones he thinks are obvious, and then to back-solve to get 1/3.

But there is a better way to implement Elga's actual problem in a form that has a simple solution. It is consistent with every detail of the actual problem, except the two days part. But you can make it last two days if you feel it necessary:

1. Put SB to sleep.
2. Flip two coins, a Quarter and a Dime.
3. If either of the coins is showing Tails, perform this procedure:
   1. Wake SB.
   2. Ask SB for her degree of belief (or credence, knowledge based probability, or whatever else) that the Quarter is currently showing Heads.
   3. Put her back to sleep with amnesia.
4. Turn the Dime over to show its other face.
5. Repeat steps 3.1 to 3.3.
6. End the experiment by waking SB.

Note that Elga's version is the same as this, if the Dime lands on Tails. And it is an equivalent problem, with a different ordering, if the Dime lands on Heads. And even though the grammatical tense of the question is different - present tense or past (possibly future?) - the subject of the question does not change during the experiment. Regardless of whether you call it credence or probability, or mean the current state or the as-flipped state, it is the same concept.

But now the solution is trivial. There are four possible states for (Quarter,Dime) in step 2, or at the start of steps 3 and 5. They are {(H,H),(H,T),(T,H),(T,T)}. Each has a credence (or probability), for a reasonable observer, of 1/4. Please note that this comes from the logic used by Halfers when they responded to Elga's modified problem. Elga himself worked backwards from the states where SB was given information that reduced the number of possible states from four to two.

And yes, the actual state changes in step 4, but the credence (or probability) distribution does not. By Halfer logic.

But whenever SB addresses the question, whether it is in step 3.2 or 5.2, she knows that the current state is not (H,H). This is new information. She can reasonably update her credence (or probability) of each remaining state from 1/4 to 1/3. That means her credence (or probability) that the Quarter is showing Heads is 1/3.

 

[JeffJo]

Again, I was assuming you would know how the problem works since there was a recent thread here. Anytime SB is awake, she has no memory of any possible previous awakenings. or future awakenings. So some (called Halfers) argue that she has no information about the coins, since she knew she would be awakened. So her credence that the Quarter landed on Heads must be has to be 1/2.

But my modeling of the actual problem shows why the Dime is relevant, and she does have new information about the pair of coins. So her credence that the Quarter is showing heads has to be updated to 1/3.

The point of my version is that her credence that it landed on Heads is the same as her credence that it is currently is showing Heads. (Or, if you go back to Zuboff's version, that she will be wakened two times.) In the other thread, the role of the Dime is fulfilled by a guaranteed awakening on Monday, and a possible one on Tuesday if the Quarter lands on Tails. It is not intuitively obvious why the day constitutes a random outcome. But my Dime is.

 

[PeterDonis]

some (called Halfers) argue that she has no information about the coins, since she knew she would be awakened.

The fact that she knew she would be awakened does not mean she has no information about the coins, period (other than the obvious information that they are fair). It means that she has no new information about the coins when she is awakened, that she did not have before she went to sleep.

So her credence that the Quarter landed on Heads must be has to be 1/2.

This is not a valid argument, however, even in the original version of the problem. She does not need new information about the coins (or the single coin, in the original version of the problem) to know that her credence that the Quarter (or the single coin in the original version) should be 1/3. She already has all the information she needs to determine that before she goes to sleep: that when she is awakened, the relevant sample space will have three possibilities (I'll state them in terms of the original version, in which she always gets awakened on Monday but only gets awakened on Tuesday if the coin came up tails): it is Monday and the coin came up heads; it is Monday and the coin came up tails; it is Tuesday and the coin came up tails. (The fourth possible pairing is ruled out, and is known to be ruled out before she goes to sleep.) Since only one of these three equally likely possibilities has the coin coming up heads, her credence for heads should be 1/3, and she knows this before she goes to sleep.

@PeroK gives the same response in the previous thread here:

https://www.physicsforums.com/threads/the-sleeping-beauty-problem.1010131/post-6574472

 

[PeterDonis]

these three equally likely possibilities

I should probably expand on this some, since another common "Halfer" response is to claim that the three possibilities are not equally likely, based on an argument something like this: if the coin comes up heads, there is only one awakening, whereas if the coin comes up tails, there are two: so each individual awakening if the coin comes up tails should only get half as much of the probability.

This is also not a valid argument, however. Ironically, the reason it is not a valid argument is precisely the fact that Halfers use to justify the previous invalid argument (the one I described in my previous post): that Sleeping Beauty learns no additional information when she awakens! The argument given in the previous paragraph implicitly assumes that she does. What is that additional information? The number of awakenings during the present run of the experiment. (See further comments below.)

Another, more "brute force" way of seeing why the above argument is wrong is to simply imagine a Monte Carlo simulation of a large number of runs of the experiment, say 1000. Since the coin is fair, we would expect it to come up heads 500 times and tails 500 times. On each of the heads runs, there is one awakening, and on each of the tails runs, there are two, for a total of 1500 awakenings, 500 of which are heads. So the fraction of awakenings at which the coin came up heads is obviously 1/3.

The Halfer argument I described above would require weighting each of the tails awakenings in the Monte Carlo simulation above only half as much as each of the heads awakenings. But on what grounds would you do that? The only possible grounds would be that each tails awakening is part of a single experimental run that contains two such awakenings, whereas each heads run is part of a single experimental run that contains just one awakening. But Sleeping Beauty does not know when she is awakened which group of runs her current run is in. All she knows is that she was awakened, and so the correct weighting is by awakenings, not by experimental runs.

Of course the Monte Carlo simulation gives another obvious way to see why 1/2 is not the right answer. Consider a 1000-run experiment in which Sleeping Beauty bets $2 on Sunday at the start of each run, and on each awakening is asked whether the coin came up heads or tails, and is paid $3 every time she answers correctly (the $3 is put in escrow until the whole experiment is over so Sleeping Beauty does not know, on any awakening, how much money she has gained or lost). Since Sleeping Beauty learns no new information on any awakening, her optimal strategy is to answer either "heads" or "tails" on every awakening. If she is a Halfer, she will have no reason to prefer the "tails" answer, whereas if she is a "Thirder", she will certainly prefer the "tails" answer. And, of course, if she answers "tails" every time she will gain $1000 (she bets a total of $2000 and is paid $3 on each of 1000 tails awakenings, and nothing on each of 500 heads awakenings), whereas if she answers "heads" every time she will lose $500 (she bets a total of $2000 and is paid $3 on each of 500 heads awakenings, and nothing on each of 1000 tails awakenings).

 

[Dale]

Consider a 1000-run experiment in which Sleeping Beauty bets

I think that the betting approach makes it obvious. But I have had discussions where people suggest all sorts of weird bets. One of the milder is to do low-stakes bets on tails. To me this is obviously wrong. The goal of the betting is to test if her belief is rational. And since the three waking scenarios are indistinguishable to her, the bets should be similarly indistinguishable.

 

[PeterDonis]

I think that the betting approach makes it obvious.

I personally use this kind of approach often to give things like "credences", "degree of belief", "probability", etc. a concrete meaning. People can argue endlessly about what words like that might or should mean, but it's hard to argue with a simple mathematical calculation that shows who wins money and who loses money.

 

[PeterDonis]

it's hard to argue with a simple mathematical calculation that shows who wins money and who loses money.

Having said that, though, even the betting approach has an interesting wrinkle in this problem.

Looking at the betting odds I gave (bet $2, pay $3 on a correct answer), a Halfer might object that I have favored the house: fair odds on a bet on a coin flip would be bet $2, pay $4, or more simply bet $1, pay $2. If we give these odds, a Halfer who answers heads whenever they are awakened would just break even (bet a total of $1000, pay $2 on each of 500 heads awakenings). If such Halfers were never told about anyone answering tails and what their payoff was, they could perfectly well conclude from the evidence they have that the odds of heads are 1/2 and not 1/3!

 

[PeterDonis]

Consider a 1000-run experiment in which Sleeping Beauty bets $2 on Sunday at the start of each run

Another possible Halfer response to this is that I have skewed things by having the bet made on Sunday, and that a fairer betting approach would be to have Sleeping Beauty bet each time she is awakened. Unfortunately for them, using the betting odds from my last post just now (bet $1, pay $2 on a correct answer), this approach is actually worse for the Halfer who always answers heads! For now, the halfer bets $1 on each of 500 heads awakenings and gets paid $2 on each of them; but they also bet $1 on each of 1000 tails awakenings and get paid nothing. Thus they have a $500 loss, whereas at these odds they just broke even on my previous betting approach.

Always answering tails on this approach also has a smaller net payoff, betting $1 on each of the 1500 awakenings and getting paid $2 on 1000 of them, for a net gain of $500, but of course this is still a better result than answering heads. So while one could argue for either of the betting approaches in terms of which is "correct", one cannot argue the fact that always answering tails beats always answering heads.

 

[JeffJo]

I had really hoped this wouldn't degrade into the same arguments between halfers and thirders. Which seldom, if ever, convince anybody firmly entrenched on the other side of that fence.

I gave what I thought was an iron-clad presentation of how the new (yes, I failed to include this word because I thought it was implied) information is provided. I used the halfer's own concept, that SB can represent a coin flip as a valid random variable. Whether they state it this way or not, their entire argument is that a day - Monday or Tuesday - cannot be represented by a valid RV, so it cannot be used to construct new information. They even have a name for this: "indexical" information. I can't claim to understand how they think it applies, since imo they have the "index" that eliminates its distinction. But they use it to merge Monday into Tuesday somewhat like how Schrodinger merged "live cat" into "dead cat."

This halfer concept also affects the so-called "betting argument," since it alters the number of times you can apply the betting paradigm. Thirders want to apply it once, or twice, depending on the coin. Halfers want to apply it exactly once, because the merged Monday-or-Tuesday concept allows only one. Once the number of times is accepted, each side's pet answer yields a balanced betting game.

So I really would like to drop the betting aspect. And in fact, betting is defined in terms of credence, not the other way around. A balanced game is a necessary result of the correct credence, not a sufficient cause to determine that credence. Which goes back to the number of bets.​

My point was to evaluate SB's credence of an existing situation that has no correlation to the timing, so the number of bets is irrelevant. What is the current state of the Quarter? In the current state, it is linked to the state of the Dime. And one combination, that initially was possible, is not possible in the current state. SB knows this, so she has current information about the combination which is different than the initial information. Viola!, new information.

I know what thirders will say about the original problem. I don't need their arguments re-hashed for me as if they know more about this problem than I do. I want to know why firm halfers think SB should have equal belief in Heads or Tails, in the current state. Or why the current state of the Quarter does not match its initial state.

 

[Dale]

Another, more "brute force" way of seeing why the above argument is wrong is to simply imagine a Monte Carlo simulation of a large number of runs of the experiment, say 1000. Since the coin is fair, we would expect it to come up heads 500 times and tails 500 times. On each of the heads runs, there is one awakening, and on each of the tails runs, there are two, for a total of 1500 awakenings, 500 of which are heads. So the fraction of awakenings at which the coin came up heads is obviously 1/3.

I did do a Monte Carlo simulation in the other recent sleeping beauty thread. I simulated 10,000 trials. That resulted in 4907 awakenings with heads and 10186 awakenings with tails. So $\hat P(head|wake)=0.325 \approx 1/3$

 

[Dale]

I gave what I thought was an iron-clad presentation

I don't know. I generally dislike additional complications in scenarios. I didn't find the inclusion of the extra random variable to be very convincing, and I agree with the conclusion. I doubt any halfer will be convinced, but I suppose we will need to wait for one to opine.

 

[PeterDonis]

I had really hoped this wouldn't degrade into the same arguments between halfers and thirders.

I haven't seen any Halfers in this thread. I was simply responding to your statements about what Halfers would argue.

 

[Moes]

After a long discussion in the previous sleeping beauty thread I still didn’t fully understand the solution to this problem. Even though I am still not clear about this I thank the people who tried to help me understand.

So I really would like to drop the betting aspect. And in fact, betting is defined in terms of credence, not the other way around. A balanced game is a necessary result of the correct credence, not a sufficient cause to determine that credence. Which goes back to the number of bets.

I agree with this. I explained the same thing in the previous thread.

My point was to evaluate SB's credence of an existing situation that has no correlation to the timing, so the number of bets is irrelevant. What is the current state of the Quarter? In the current state, it is linked to the state of the Dime. And one combination, that initially was possible, is not possible in the current state. SB knows this, so she has current information about the combination which is different than the initial information. Viola!, new information.

I find your variation of the problem interesting but still don’t see how it answers the “no new information” argument. If the quarter landed heads she knows she will only wake up at the time where the dime is facing tails. So she knows she would only be able to think about the state of the quarter when the dime is facing tails. Knowing that this is presently the time that she is thinking about it shouldn‘t give her information about how the quarter landed.

I have a weird way of thinking about this: Knowing that you are thinking at a time where it’s possible for you to be thinking should never be surprising or give you any information.

Tell me if you think I’m crazy.

 

[PeterDonis]

I gave what I thought was an iron-clad presentation of how the new (yes, I failed to include this word because I thought it was implied) information is provided.

There is no new information provided on any of the awakenings. Every time Sleeping Beauty awakens, on either the original formulation or your new one, she has the same information she had at the start of the experiment. And that information is already sufficient to show that her answer to the question she is posed should be 1/3. The experimenters have information when Sleeping Beauty is awakened that they did not have at the start of the experiment, but none of their new information is passed on to Sleeping Beauty.

whenever SB addresses the question, whether it is in step 3.2 or 5.2, she knows that the current state is not (H,H). This is new information.

No, she doesn't, because she already knows, at the start of the experiment, that she will never be awakened if the current state is (H, H), so she can calculate her credence with that knowledge at the start of the experiment.

 

[PeterDonis]

Halfers want to apply it exactly once, because the merged Monday-or-Tuesday concept allows only one.

How would this work if the coin comes up tails? If the coin comes up tails, Sleeping Beauty is awakened twice, on Monday and Tuesday, and there can't be any difference between the two awakenings. So whatever betting scenario she is presented with on Monday, she has to be presented with the same betting scenario on Tuesday.

 

[PeterDonis]

she knows she would only be able to think about the state of the quarter when the dime is facing tails. Knowing that this is presently the time that she is thinking about it shouldn‘t give her information about how the quarter landed.

Your analysis of why there is no new information is correct (it's basically the same as the analysis I gave in post #14). The question is, do you agree that Sleeping Beauty's credence should be 1/3? I have explained how she can calculate that with the information she has at the start of the experiment, before she is put to sleep.

 

[PeterDonis]

A balanced game is a necessary result of the correct credence, not a sufficient cause to determine that credence.

Which means that, if the Halfers are right that 1/2 is the correct credence, it should not be possible for a Thirder who answers tails to beat a Halfer who answers heads in the betting games I described with the correct "coin flip" odds (bet $1, get paid $2 on a correct answer), since a Halfer would say that those games are balanced. So the fact that a Thirder who always answers tails does beat a Halfer who always answers heads at those games must mean, by your argument, that those games are not balanced--the odds would have to be changed to make them balanced, and that would mean the correct credence, which determines the balanced game odds, could not be 1/2.

 

[Dale]

After a long discussion in the previous sleeping beauty thread I still didn’t fully understand the solution to this problem.

When I am confused about something in probability, I always do a Monte Carlo simulation to resolve it. A simulation cuts through all of the BS, and you can directly code the original scenario without needing any alterations to make the point.

I have already done it and it clearly showed $P(heads| awake)=1/3$. I recommend you write your own and see.

 

[Moes]

When I am confused about something in probability, I always do a Monte Carlo simulation to resolve it. A simulation cuts through all of the BS, and you can directly code the original scenario without needing any alterations to make the point.

I have already done it and it clearly showed $P(heads| awake)=1/3$. I recommend you write your own and see.

I don’t know enough about simulations to know the exact information they give you. And there are still a lot of halfers despite the results of the simulation. And anyways I really want to fully understand why either position is correct the facts are not so important to me. But thanks anyway.

Your analysis of why there is no new information is correct (it's basically the same as the analysis I gave in post #14). The question is, do you agree that Sleeping Beauty's credence should be 1/3? I have explained how she can calculate that with the information she has at the start of the experiment, before she is put to sleep.

I didn’t understand your explanation. Before the experiment starts her credence of the coin is 1/2. Why should thinking about a future event that will for sure take place change that credence? I understand that when she thinks “When I wake up ….” 2/3 of the “When”s the coin will have landed tails but this shouldn’t change the probability that when she does wake up the coin will have landed tails or heads it’s 50/50.

 

[PeterDonis]

Before the experiment starts her credence of the coin is 1/2.

No, it isn't.

To be more precise: if she were asked, before the experiment starts, "what is your credence that the coin will come up heads when it is flipped?", her answer would be 1/2. But that is not the question she is asked. The question she is asked is, "Given that you have just been awakened, what is your credence that the coin came up heads?" That is a different question and has a different answer, 1/3. And it is a question and answer she can already calculate given the information she has at the start of the experiment.

I understand that when she thinks “When I wake up ….” 2/3 of the “When”s the coin will have landed tails but this shouldn’t change the probability that when she does wake up the coin will have landed tails or heads it’s 50/50.

There is no such thing as "the" probability the coin lands heads or tails. There are only conditional probabilities, and the conditional probability "given that the coin was just flipped, it landed heads" (1/2) is different from the conditional probability "given that Sleeping Beauty was just awakened, the coin landed heads" (1/3). And the latter is the conditional probability that is relevant for answering the question Sleeping Beauty is actually asked.

 

[Dale]

I don’t know enough about simulations to know the exact information they give you.

They are like actually doing the scenario 10000 times and seeing what happens.

And there are still a lot of halfers despite the results of the simulation.

Have any of them actually done a simulation? I haven’t seen it if they have.

And anyways I really want to fully understand why either position is correct the facts are not so important to me.

The problem is that you have already been given many explanations why each position is correct and you are not well-versed enough to see the right or wrong in those explanations (including your own).

The simulation makes it clear which position is correct. Then you can focus on understanding the arguments for or finding the flaws in the opposing arguments. Essentially, this is like working a physics problem two good-seeming but contradictory ways, but checking the answer key. Then you can go back with a critical eye and spot the flaw in the incorrect approach.

 

[Dale]

Before the experiment starts her credence of the coin is 1/2. Why should thinking about a future event that will for sure take place change that credence?

Before proceeding, please post the scientific reference you are using for your understanding of the term “credence”.

 

[JeffJo]

I find your variation of the problem interesting but still don’t see how it answers the “no new information” argument.

In the prior, the equivalent of "Sunday Night" in the original problem, the sample space comprises four outcomes: {(H,H), (H,T), (T,H), (T,T)}. Each event that contains exactly one of them has prior probability 1/4.

When SB is asked for her credence about the current state, one of those outcomes, (H,H), is no longer possible. The other three remain possible. This is a classic example of "new information," so I fail to see how you can't see it as such.

Every time Sleeping Beauty awakens, on either the original formulation or your new one, she has the same information she had at the start of the experiment.

She has the same information about how the experiment is run, which seems to be what you are describing. That isn't what "new information"applies to. a She has new information about the possible state of the coins. In the prior, four combinations are possible. In her current state, only three are.

No, she doesn't, because she already knows, at the start of the experiment, that she will never be awakened if the current state is (H, H), so she can calculate her credence with that knowledge at the start of the experiment.

But she knows that (H,H) is a state that is possible. The fact that she will not be awake if it occurs does not alter reality.

How would this work if the coin comes up tails? If the coin comes up tails, Sleeping Beauty is awakened twice, on Monday and Tuesday, and there can't be any difference between the two awakenings. So whatever betting scenario she is presented with on Monday, she has to be presented with the same betting scenario on Tuesday.

You will have to ask a halfer.

Your analysis of why there is no new information is correct (it's basically the same as the analysis I gave in post #14). The question is, do you agree that Sleeping Beauty's credence should be 1/3?

That is what I said the answer was in the original post.

 

[Moes]

Before proceeding, please post the scientific reference you are using for your understanding of the term “credence”.

I ended up unsure whether our argument was about the definition . The arguments I was having with others didn’t seem to have to do with that. So I think we should leave that out. I believe I agree with the references you gave . My point was really just like JeffJo said:

So I really would like to drop the betting aspect. And in fact, betting is defined in terms of credence, not the other way around. A balanced game is a necessary result of the correct credence, not a sufficient cause to determine that credence. Which goes back to the number of bets.

 

[PeterDonis]

In the prior, the equivalent of "Sunday Night" in the original problem, the sample space comprises four outcomes: {(H,H), (H,T), (T,H), (T,T)}. Each event that contains exactly one of them has prior probability 1/4.

That's not the relevant "prior" because Sleeping Beauty already knows on Sunday Night that she will not be awakened at all in the (H, H) case, so that case drops out of any calculation of probabilities conditioned on her being awakened. And she knows that the question she is going to be asked is about a probability conditioned on her being awakened (since she knows she will only get asked the question on being awakened).

When SB is asked for her credence about the current state, one of those outcomes, (H,H), is no longer possible.

But she already knows that this will be the case on Sunday Night. So on Sunday Night she already has all the same information she will have when she is awakened. She does not learn any new information when she is awakened that she did not already know. The only difference is that on Sunday Night she is pre-computing the answer to the question she will be asked, whereas when she is awakened she is giving her pre-computed answer to the question she is being asked. That difference does not change any probabilities.

But she knows that (H,H) is a state that is possible.

And she also knows that she will not be awakened at all in this state, so she knows this state is irrelevant for computing probabilities conditional on her being awakened, which are the relevant probabilities for questions she will be asked when she is awakened.

 

[PeterDonis]

That is what I said the answer was in the original post.

The question of mine that you quoted wasn't addressed to you, it was addressed to @Moes.

 

[Moes]

"given that Sleeping Beauty was just awakened, the coin landed heads" (1/3). And the latter is the conditional probability that is relevant for answering the question Sleeping Beauty is actually asked.

I think my problem is that when you say "given that Sleeping Beauty was just awakened” you are already leading to an answer of tails since that “given” is double in the event of tails then it is in heads. This is before you are even getting to what the probability of the coin actually landing tails is.

 

[PeterDonis]

You will have to ask a halfer.

No, I'm asking you, because you made a claim about what a halfer would say. If you're not going to back up that claim with details, you should not have made it in the first place.

 

[PeterDonis]

I think my problem is that when you say "given that Sleeping Beauty was just awakened” you are already leading to an answer of tails since that “given” is double in the event of tails then it is in heads.

Yes, that is a crucial factor in computing the conditional probability of heads given that she was awakened. That's the point.

This is before you are even getting to what the probability of the coin actually landing tails is.

No, it's after knowing that the probability of the coin landing heads or tails, given that it was flipped, is 1/2.

Again, there is no such thing as "the" probability of the coin landing tails. There are only conditional probabilities. The 1/2 is the probability of the coin landing tails (or heads) conditioned on the fact that it was flipped. But, as I've already said, that is not the correct conditional probability for the question Sleeping Beauty is actually asked.

 

[Moes]

Again, there is no such thing as "the" probability of the coin landing tails. There are only conditional probabilities. The 1/2 is the probability of the coin landing tails (or heads) conditioned on the fact that it was flipped. But, as I've already said, that is not the correct conditional probability for the question Sleeping Beauty is actually asked.

I think I understand this point. So the question she is asked is what is her credence of heads conditioned on the fact that the coin was flipped and that she is now ( at a time where it’s possible for her to be thinking about this) awake. Whether the coin landed heads or tails she would now be awake so I don’t see how this condition could change the probability.

I’m not sure why I’m not understanding your explanation. If you have anything more to explain please do.

 

[PeterDonis]

the question she is asked is what is her credence of heads conditioned on the fact that the coin was flipped and that she is now ( at a time where it’s possible for her to be thinking about this) awake.

Yes; I did not include "the coin was flipped" in the conditions before, but adding it to the conditions is fine and does not change anything I've said.

Whether the coin landed heads or tails she would now be awake

But whether the coin landed heads or tails does change the number of times she is awakened. And since when she is awakened she does not know how many times she will be or has been awakened, she has to include all the possible awakenings when she computes the conditional probability of heads given that she has been awakened.

I don’t see how this condition could change the probability.

See above.

 

[PeterDonis]

I don’t see how this condition could change the probability.

The phrase "change the probability" is misleading. No probability "changes". There are simply different conditional probabilities based on different conditions. Conditioning on Sleeping Beauty being awakened and the coin having been flipped is different from conditioning on just the coin having been flipped.

 

[Dale]

I believe I agree with the references you gave

Then the betting follows.

There has been some comments that the betting doesn’t cause the credence, which is true: the betting measures the credence. And since Sleeping Beauty is described as rational she won’t take bets that she expects to lose and it is that rationality which then constrains her credence. Because she is rational we can infer her credence from the expected value of a wager.

 

[JeffJo]

That's not the relevant "prior" because Sleeping Beauty already knows on Sunday Night that she will not be awakened at all in the (H, H) case, so that case drops out of any calculation of probabilities conditioned on her being awakened. And she knows that the question she is going to be asked is about a probability conditioned on her being awakened (since she knows she will only get asked the question on being awakened).

One of the interesting aspects of Probability is that there can be many ways to create a sample space. If I roll two six-sided dice, I can create a sample space with 36 pair-combinations, or 11 sums. It's just easier to assign probabilities to the pair-combinations.

You can claim that "a relevant prior" is the state of the coin(s) that SB can predict, on Sunday Night, will exist when she awakes. That's completely impractical, since it is also the posterior she needs to use. So of course there is no "new information."

But there is no such thing as "the relevant sample space." You are using circular reasoning, by claiming I can only use the same space for both prior and posterior. And quite wrong.

My prior sample space, which I find to be more relevant than yours, is {(H,H), (H,T), (T,H), (H,H)}. The probability for each one-outcome event is 1/4.

• This is what the experiment moderator sees when he decides whether or not to wake SB.
• The resultant event space exists whether or not he wakes her.
• It continues to exist whether or not he wakes her.
• By waking SB, he does not create a sample space. He makes her a conditional observer of the one that exists.
• When she becomes an observer, she can deduce that (H,H) is no longer a possibility.
• THIS IS NEW INFORMATION.

Please note that SB knows, before being put to sleep, how each part of this process can go. She knows the coins will be tossed. Se knows that there are four possible combinations the moderator will look at. She knows what will happen in each contingency. She knows she will be asked a question in only three of the four.

But she already knows that this will be the case on Sunday Night.

Yep. What has that to do with whether, in my prior (which has the advantage that the probabilities are not controversial), (HH) was a possibility and now it isn't?

And she also knows that she will not be awakened at all in this state, so she knows this state is irrelevant for computing probabilities conditional on her being awakened, which are the relevant probabilities for questions she will be asked when she is awakened.

But the point you seem to overlook is that the state exists whether or not she is made an observer.

So change it. Wake her before and after you reverse the dime (or on both Monday and Tuesday in the original). If both coins are showing Heads in my version (or the coin landed Heads and it is Tuesday), ask her whether she likes Gone With the Wind more than The Wizard of Oz. Otherwise, ask her for her credence in Heads. But now she has what I hope you would call "new information." The prior is the one I called relevant, and the posterior is the one you said was the relevant prior.

She can say her confidence in Heads is 1/3.

But now, why does it matter if you lied, and had no intention of waking her if you already knew her preference about movies? The state that determines what type of - or even if - an observer she becomes is a valid prior.

No, I'm asking you, because you made a claim about what a halfer would say. If you're not going to back up that claim with details, you should not have made it in the first place.

And I'm saying that, just like I can't understand why you want to call what is clearly as posterior "the relevant prior," I have never seen it justified. If you want to see an example of it, google for "double halfers."

 

[Dale]

I am also not convinced by the “no new information” claims. I would like to see an actual calculation from information theory to support that claim. I did such a calculation previously and got 1 bit of information, which seems too high. So I am not confident in my result, but I really don’t think that the 0 information claim is convincing.