Undergrad The Sleeping Beauty Problem: Any halfers here?

[Stephen Tashi]

We either have to be told to assume probabilities, or find a way to apply the PoI.

In mathematical problems, certain probabilities are given and we are asked to solve for others. If you want to say we are "assuming" probabilities, when we take the given information for granted, that is correct as far as I'm concerned.

Wm and Wt are ambiguous events since they cannot be evaluated in the posterior.

Their meaning is not ambiguous. They may not useful in a posterior distribution, but there is nothing in the statement of the problem that requires that the events in the experiment be useful in some posterior distribution. In fact, the events described in the experiment provide no information about what happens "today" unless you hypothesize some stochastic process that is selecting an event from the experiment that will apply to "today".

One way to tell that you are using inconsistent concepts, is that the usual halfer claim is that "SB knows she will be wakened." She does not know that this event "a" will happen, since today might be Tuesday.

We are only asked to compute probabilities in the situations when SB is awakened.

The issue I keep pointing out to you, is that an awake SB is seeing only a portion of the overall experiment. She must use a probability space that applies to that portion only.

The problem says SB knows the conditions of the experiment. It would be irrational for her not include them in her deliberations.

TODAY has a value in {Mon, Tue}. Since both look the same to SB, she cannot use different probabilities for them. The PoI applies the say way.

Is that supposed to be an application of the "strong" PoI or the "weak" PoI?

They are independent, so the prior probability for each combination in {(H,Mon),(H,Tue),(T,Mon),(T,Tue)} is 1/4.

I think you mean that they are mutually exclusive events, not independent events.

You take a typical "thirder" approach , which I myself don't mind. However, the problem doesn't mention any stochastic process that randomly selects one those events to be "today". Nor does it say that Sleeping Beauty is required to think about a prior distribution for such a process.

As I said before, a mathematical solution to a proability problem can be checked without knowing how it was derived. You think Sleeping Beauty must assign equal probabilities to the events "today" is Monday and "today" is Tuesday before the experiment begins and then update that prior distribution when she is awakened. Just because the "halfer" solution violates an assumption you make in deriving the "thirder" solution does not show the "halfer" solution is incorrect.

The "halfer" solution does contradict the procedure you use to compute the "thirder" answer. If you reject the "halfer" solution on those grounds then the criteria you are using for correctness is that Sleeping Beauty is required to engage in certain thought processes - apparently she is required to think like a Bayesian.. (Good, I have Bayesian tendencies). Under the assumption that Sleeping Beauty must execute certain algorithms and compute P(H|a) using a certain repertoire of assumptions, you might be correct that the unique answer is 1/3. All I'm saying is that the information given in the problem (which does not identify Bayesian thinking as the only rational conduct for SB) is not sufficient to determine a unique solution.

There is, if we apply it to the (as admitted by halfers) equivalent problem where the coin isn't flipped until Tuesday Morning. How does the coin know that it has to land Tails with probability 2/3?

How does the coin know that needs go back in the past and change the probability that it landed heads to 1/3?

Let's try this again. Four equally rational women (use your definition for "rational") participate in an OP-like experiment using the same coin. There are only two possible differences from the OP: the circumstances under which they are left asleep are different for each, (H,Mon), (H,Tue), (T, Mon), and (T,Tue), respectively; and they are asked about the coin result in that set of circumstances, not necessarily "Heads".

Question that you seem to be avoiding: Should each arrive at the same conclusion about the solution to the problem, whether that be "1/2", "1/3", "ambiguous", or something else? All I seek is a "yes" or a "no," not equivocation or what that answer is.

You offer so many purportedly equivalent problems, I can't keep track of them. From the description above , the experiment is not clear. And if your problem is ill-posed there need be no yes or answer.

I am taking a serious look at one of your proposed problems. It will take time.

 

[Dale]

That requires that we define a bet for the Sleeping Beauty problem. What is it?

E is "the coin landed heads". The bet is clearly defined in the quote, and such a wager is implied every single time Beauty is asked about her credence. This is not unclear or ambiguous in any way, and the rational answer is 1/3.

For example suppose the bet is that ...

Did you quote the definition without even reading it? There is no need to suppose. The bet is clearly defined.

 

[Stephen Tashi]

E is "the coin landed heads". The bet is clearly defined in the quote, and such a wager is implied every time Beauty is asked about her credence. This is not unclear or ambiguous in any way and the rational answer is 1/3.

I think you mean that the bet is for the owner of the bet to receive $1 if the coin landed heads. (Of course you always say I'm putting words in your mouth.)

That doesn't satisfy the definition of credence because the definition of credence deals with the decision about buying a single bet for a known cost. It doesn't cover the case where the decision to buy one bet would force the person to buy another different bet.

Being a slave to the consistency of rationality, if Sleeping Beauty decided to buy the bet on heads for X dollars, she would have to buy the bet for X dollars every time it was offered since she can't detect any difference in conditions that affect the bets. The net price she pays for her decision is stochastic. It might be X and it might be 2X.

We can make a nice quasi-"thirder" argument based on those ideas, without getting into any controversy about the value of P(H|a).

The expected payoff of the bets is (1/2)$ The expected cost fo Sleeping Beauty if she always buys the bet for X dollars is X + (1/2)X. The appropriate value for bets with expected return of (1/2)$ is (1/2)$. So Sleeping Beauty should evaluate the bet(s) as being worth (1/2)$ = X + (1/2)X. So X = (1/3)$ and Sleeping Beauty's credence is 1/3.

However, this is not her credence in the event " The coin landed heads" because the bet isn't simply "If the coin landed heads you are paid $1". Instead, the (1/3)$ is Sleeping Beauty's evaluation of the wager: "If the coin landed heads you are paid $1 and your cost is what you offered for the bet and if the coin did not land heads your cost is twice what you offered for the bet."

 

[Dale]

That doesn't satisfy the definition of credence

This is getting frustrating. Read the definition. The bet in the definition satisfies the definition by definition.

if Sleeping Beauty decided to buy the bet on heads for X dollars, she would have to buy the bet for X dollars every time it was offered since she can't detect any difference in conditions that affect the bets

Precisely!

However, this is not her credence in the event " The coin landed heads"

Yes, it is, by definition. It is her credence in heads every time that she is asked.

Read the definition! If you are still confused then read the commentary as well.

 

[Stephen Tashi]

Yes, it is, by definition. It is her credence in heads every time that she is asked.

No, it isn't.

You are considering only part of event that determines the bet and computing the payoff from the bet as if it only depends on that part of the event.Consider the bet: A fair coin is tossed. If it lands heads then you get $1. If it lands tails then we roll a fair die and you are charged $1 unless the die lands as a six.

The fair price for that bet is not the credence for the event "The coin lands heads".

The bet in Sleeping Beauty's case amounts to: "We toss a fair coin. If the coin lands heads you get $1. If coin lands tails you are charged whatever you paid for this bet" .

Sleeping Beauty can evaluate the bet before the experiment starts and get the same value of credence. The evaluation doesn't depend on any estimate of a posterior probability that the coin landed heads after she is awakened.

 

[stevendaryl]

Consider the bet: A fair coin is tossed. If it lands heads then you get $1. If it lands tails then we roll a fair die and you are charged $1 unless the die lands as a six.

The fair price for that bet is not the credence for the event "The coin lands heads".

That's not quite in the spirit of using bets to determine probability, but maybe it can be tweaked into an example.

What I would say--and there is probably some rigorous definitive definition out there already--is that for a particular kind of game such that there are two possible outcomes---you either win $1 or win nothing---then the subjective probability of winning the game is the amount of money (in fractions of a dollar) you would be willing to pay for the chance to play the game.

So does Sleeping Beauty count as this type of game, or not? Each betting occurrence seems to be an example. However, you could argue that since SB doesn't learn anything upon being awakened that she didn't know at the beginning, she may as well have placed her bet at the beginning, with the rules:

1. If she bet on tails and the result was tails, then she wins $2.
2. If she bet on heads and the result was heads, then she wins $1.
3. If she bet on tails and the result was heads, she loses $1.
4. If she bet on heads and the result was tails, she loses $2.

This way of putting it, the compound game is certainly not of the form that allows you to compute probability based on how much you would pay for the chance to play. The amount you have to bet and the amount you stand to win are not fixed.

So it's only the individual bets made on each day that has the structure of a bet that can determine probability.

 

[Ken G]

I'm coming into this late, but it seems very clear to me the correct answer is 1/3, on the simple grounds that Sleeping Beauty knows that if she guesses "the coin landed heads" every time she is awakened, and if the experiment is repeated every week for a year, then she will clearly have been correct 1/3 of the time, and none of those events will seem any different to her. So that's 1/3 credence, and I can see no other meaningful way to define the concept of "credence." I think what some people may be missing is that the experimenters can also ask her "what is your credence that today is Monday," and the answer to that will be 2/3. Hence, her credence that the coin was a heads equals her credence it is Monday, times her credence that the coin came up heads if it is Monday, plus her credence that it is Tuesday times her credence that the coin was heads given that it is Tuesday (which is zero). So the full credence she should have is
2/3 * 1/2 + 2/3 * 0 = 1/3.
The point is, when you analyze information, you don't just look at what you know, you also look at what all the hypothetical possibilities are and what you know about the likelihood of each hypothetical possibility. You must deal in hypotheticals, by which I mean information that is not certain-- not just information that is certain. As such, the day it is must be considered a hypothetical that has its own likelihood and must be included in the information being analyzed.

Hence, if you ask SB what her "heads" credence is right before she goes to sleep Sunday night, she'll say 1/2 (and be right half the time), and if you ask her again when you wake her up, the answer changes to 1/3. The question the halfers ask is, what new information does she have when she is awakened? The answer is simple-- she now knows there is a 2/3 chance it is Monday and a 1/3 chance it is Tuesday! That is clearly correct, is it not? That's new information, that comes the moment she is awakened.

 

[Dale]

The bet in Sleeping Beauty's case amounts to: "We toss a fair coin. If the coin lands heads you get $1. If coin lands tails you are charged whatever you paid for this bet"

Read the definition. This is not the defined bet.

 

[Buzz Bloom]

I am a halfer.

She will know that she be will awakened on Tuesday only 1/4 of the time. The coin must be tails, and in that case it will be Tuesday 1/2 the time.
If it is Monday, the coin would have been heads 2/3 of the time. Therefore 1/2 the time (2/3 × 3/4) it will be both Monday and the coin heads.

 

[stevendaryl]

I am a halfer.

She will know that she be awakened on Tuesday only 1/4 of the time. The coin must be tails, and in that case it will be Tuesday 1/2 the time.
If it is Monday, the coin would have been heads 2/3 of the time. Therefore 1/2 the time it will be both Monday and the coin heads.

Where are you getting that from? If you do the experiment N times, then on the average:

• There will be N/2 awakenings on Monday when the result was Heads.
• There will be N/2 awakenings on Monday when the result was Tails.
• There will be N/2 awakenings on Tuesday when the result was Tails.

So on the average, there will be 3N/2 awakenings, and of those, N/2 will be when the coin result was Heads. So the relative frequency of heads, given that she is awake is 1/3.

 

[Ken G]

What's more, the number of times she is interviewed on Monday is N, and the number of times she is interviewed on Tuesday is N/2, so clearly she should reckon the chances that it is Monday is 2/3. It seems to me that all the halfers get this calculation wrong.

 

[Stephen Tashi]

This way of putting it, the compound game is certainly not of the form that allows you to compute probability based on how much you would pay for the chance to play. The amount you have to bet and the amount you stand to win are not fixed.

I agree. If Beauty is rational, she realizes that her purchase of a bet on on particular awakening has consequences for any other awakening that happens in the experiment. The definition of credence I quoted assumes a bet where the payoff from the event E is 1 unit of utility.and an buying the bet at price X doesn't affect the payoff.

A web search this morning turns up other people, presumably more experienced in studying credence than me, who point out this complication of the Sleeping Beauty problem. For example:

https://philpapers.org/archive/YAMLSB

 

[stevendaryl]

I agree. If Beauty is rational, she realizes that her purchase of a bet on on particular awakening has consequences for any other awakening that happens in the experiment.

Her bet doesn't have consequences beyond the particular awakening. She's not forced to bet the same way every time she is awakened, it's just that it's in her best interest to do so. I don't see how that's any different from any other type of bet: if there's a unique best way to bet, then you'll bet the same way every time you're in that circumstance. That doesn't mean that one of your bets forces you to make the other bets in any particular way.

The thing that's odd about SB is that the coin toss has consequences beyond being the event that she's betting on.

 

[Stephen Tashi]

Her bet doesn't have consequences beyond the particular awakening. She's not forced to bet the same way every time she is awakened, it's just that it's in her best interest to do so.

Isn't she forced to price the bet the same way every time if the problem stipulates that Sleeping Beauty is rational? Can we beat consistency with some sort of "mixed" pricing strategy where Sleeping Beauty prices the bet by tossing a die or a fair coin of her own?

 

[Ken G]

To all "halfies," I pose this question: what odds should SB take, when awakened, that today is Monday, and what odds should she take that today is Tuesday?

 

[Stephen Tashi]

What's more, the number of times she is interviewed on Monday is N, and the number of times she is interviewed on Tuesday is N/2, so clearly she should reckon the chances that it is Monday is 2/3. It seems to me that all the halfers get this calculation wrong.

In this thread, I think some "thirders" and "halfer" claim calculate credence by first calculating P(coin laded heads | SB is awakened) objectively, using only information in the problem. From that objectively calculated probability, they obtain SB credence since a rational SB does the same calculation.

Other "thirders" and "halfers" arrive at SB's credence by asserting Sleeping Beauty must reason in certain way and add her own subjective probabilities (via the Principle of Indifference) to the given information.

The following remarks only concern the question of whether P(coin landed heads| Sleeping Beauty is awakened) can be objectively calculated. The value denoted by "P(coin landed heads | Sleeping Beauty is awakened:" is not a well defined probability unless there is some stochastic process implicitly or explicitly specified in the given information that tells us how to stochastically pick the situation in the experiment when she is awakened. One natural way to specify such a process is run the experiment a lot of times and then randomly select a situation where she was awakened from the records of the experiments, giving each an equal probability of selected. Using that method, it is correct that there are (probably) about twice as many instances where she was awakened when the coin landed tails as when the coin landed heads. That method supports the "thirder" answer. However, my (perhaps very technical objection) is that the information in the problem does not specify any method for picking what situation occurs when Sleeping Beauty is awakened. One rebuttal to my objection is everybody knows that you are supposed assume that "thirder" sort of method is what is used. This becomes a debate about conventions of interpreting probability word-problems. Another rebuttal to my objection is that a rational Sleeping Beauty would assume that the "thirder" type of method is used. That rebuttal is not relevant to computing the objective value of P(coin landed heads | Sleeping Beauty) is awakened. Instead, it Is relevant to the different question of how P(coin landed heads | Sleeping Beauty is awakened). is calculated subjectively.

In regard to the interpretation of word problems, the information does tell us that we must consider "whenever" Sleeping Beauty is awakened. So a stochastic process that selects the situation that applies when "Sleeping Beauty is awakened" cannot leave out any of the situations. However, we aren't required by the given information to pick the situation that applies from any particular distribution. From a Bayesian point of view, it is natural to assume a prior distribution where each situation has an equal probability of being chosen.

 

[Ken G]

However, my (perhaps very technical objection) is that the information in the problem does not specify any method for picking what situation occurs when Sleeping Beauty is awakened.

Thanks for explaining further, but I don't understand what you mean by a "method for picking". SB is in a real situation, and she is really offered odds to make various bets. I think the issue of the coin toss is causing some confusion, so that's why I asked a different question-- what odds should she take that it is Monday? Regardless of what reasoning you use, this is a perfectly straightforward question, since she could really be offered various odds, and she could actually make or lose money in the long run by accepting or rejecting those odds. So what would you say? What odds should she take that it is Monday? You are allowed to assume any "method for picking" that you like, as long as the experimenters actually flip a coin, actually waken SB, actually apply the amnesia elixir, and actually offer the odds in question that it is Monday. I can't see any method you could imagine where those odds are not simply 2/3, in the sense that SB will certainly make money in the long run by accepting any odds higher than 3-to-2 in favor, and receiving payoff if it is indeed Monday. So we need to start from a place where we can agree on this, before we even consider the coin toss.

You see, as soon as one raises the issue of the odds of what day it is, the halfer response is immediately refuted, because if it is Monday, the coin toss has a 50-50 chance, and if it is Tuesday, the coin toss was tails. Ergo, the only way you can be a "halfer" is if you think you know it can't be Tuesday, or else your reasoning is inconsistent.

 

[Buzz Bloom]

I am a halfer.

I now realize I made an error, and am now a thirder. My error was a misreading of the problem statement to be saying if tails occurs, then a choice is made between a Monday or a Tuesday interview. For N flips of the coin this leads to the expected event distribution:

Heads & Awake Monday - Expected to occur N/2 times
Heads & Awake Tuesday - Expected to occur zero times
Tails & Awake Monday - Expected to occur N/4 times
Tails & Awake Tuesday - Expected to occur N/4 times​

I wonder if all halfers make this mistake.

The correct interpretation is that tails leads to both a Monday and Tuesday Interview. For N flips of the coin this leads to the expected event distribution:

Heads & Awake Monday - Expected to occur N/2 times
Heads & Awake Tuesday - Expected to occur zero times
Tails & Awake Monday - Expected to occur N/2 times
Tails & Awake Tuesday - Expected to occur N/2 times​

Thus interviews with heads occurs only 1/3 of the time.

 

[PeterDonis]

This is getting frustrating.

I can understand that reaction, and I don't want to rehash everything that's already been said, but I think I observed before that IMO the Sleeping Beauty problem is not a good illustration of the concept of "credence", and @Stephen Tashi is referring to one reason why: the amount that Beauty will end up wagering on the coin flip depends on the result of the coin flip (because she ends up wagering once if the coin is heads, but twice if the coin is tails, and each wager is identical because she has the same information each time). So the odds that she is willing to accept on heads are affected, not just by the odds that the coin will turn up heads, but by the weird structure of the experiment that makes the payoffs skewed. In your formulation, the skew is the ratio P(A|H) / P(A|T) = 1/2.

I understand that, in the references you have given, the concept of "credence" is defined in such a way that you can indeed argue that this weird skewed structure still leads to a valid answer of 1/3 to the question "what is your credence now that the coin turned up heads?" But I am also sympathetic to reactions like that of @Stephen Tashi (since I have a similar reaction), which are basically along the lines of: you're setting up a weird, skewed experiment and then trying to claim that the answer 1/3 is just "the credence that the coin came up heads" instead of "the credence that the coin came up heads given that you've skewed the payoffs". Yes, if you actually put me into this experiment, and you explained to me exactly what you meant by "credence that the coin came up heads", I would have to insist on 2:1 odds on heads (i.e., P(Heads) = 1/3). But I would still feel like you were abusing language by phrasing the question that way.

 

[PeterDonis]

The value denoted by "P(coin landed heads | Sleeping Beauty is awakened:" is not a well defined probability unless there is some stochastic process implicitly or explicitly specified in the given information that tells us how to stochastically pick the situation in the experiment when she is awakened.

As I just posted in response to @Dale , I understand this reaction and where it is coming from. However, I would point out, in defense of his position, that there is no "stochastic" process involved at all in the experiment, except for the initial coin flip itself. Once the coin flip is made, everything is perfectly deterministic. The thing that makes the situation weird, and gives rise to intuitive objections to accepting 1/3 as "the credence that the coin came up heads", is, as I said in my post just now, that the structure of the experiment skews the total payoffs by making the total amount wagered dependent on the results of the coin flip--if the coin comes up heads, Beauty only ends up wagering once, while if it comes up tails, she ends up wagering twice, and each wager must be identical since she has the same information each time (because of the amnesia drug). This is not a "stochastic" result; it's just an odd but deterministic feature of the experimental setup.

So, as I've said before and repeated in my last post, if I were actually put into this experiment, and it were explained to me exactly what the experimenter meant by "credence that the coin came up heads" (which is the odds I will accept on a wager that the coin came up heads, given that the number of times I will end up making that wager depends on the result of the coin flip as just described), I would have to insist on 2:1 odds on heads, to make up for the skewed payoff structure. I might still object to using the language "credence that the coin came up heads" to describe this result, but that's a question of words, not results: once I know what the experimenter means by that (even if I would not use the same ordinary language to convey that meaning), I know how I have to bet.

So my question to you would be, if you were put into the experiment under the same conditions, with the same knowledge of what the experimenter meant by "credence that the coin came up heads" (even if you didn't agree with his choice of words), how would you bet?

 

[Buzz Bloom]

she ends up wagering once if the coin is heads, but twice if the coin is tails

Hi Peter:

I have not read all 500 posts in this thread, and I have missed any discussion of wagering. I do not understand how wagering is part of the problem. I do not see the connection between wagering and credence. Can you explain this?

Regards,
Buzz

 

[Stephen Tashi]

Thanks for explaining further, but I don't understand what you mean by a "method for picking". SB is in a real situation, and she is really offered odds to make various bets.

I agree that the question posed n the Sleeping Beauty Problem concerns "credence" and this leads to questions about a betting strategy.

The question of whether P(heads | SB awakened) can be objectively calculated is a side issue, but it's one that comes up again and again. I'm saying that calculating P(heads| SB) awakened from the information in the problem objectively requires knowing (or deducing) a particular probabiiity distribution F on the situations { (heads, Monday, awakened) , (tails, Monday, awakened), (tails, Tuesday, awakened)}. The distribution F defines the "method of picking" the situation when SB is awakened. The distribution F implies a particular value for P(heads | SB is awakened) by using objective calculations for conditional probabilities.

A "thirder" choice for F is to assign each situation an equal probability. That is information not given in the problem, although it is a plausible Bayesian assumption.

Suppose the process for picking the situation when SB is awakened is to run the experiment a large number of times and select a situation when SB is awakened at random from the situations that happened in the experiments, giving each such situation an equal probability of being selected. This method does imply F is a uniform distribution on the situations. However, the information in the problem does not specify this particular method for picking the situation.

When we get into betting strategies, it is relevant to run the experiment a large number of times and consider how each situation that arises in the experiment affects the strategy. We have to do this because the problem says that the betting occurs each time Sleeping Beauty is awakened. So calculating the result of a strategy by using probability theory must use probabilities that are consistent with the thought that each situation that arose in a large number of experiment is considered exactly once .

 

[PeterDonis]

I do not see the connection between wagering and credence.

See this Wikipedia article (which is linked to from the one on the Sleeping Beauty problem that is linked to in the OP of this thread):

https://en.wikipedia.org/wiki/Credence_(statistics)

 

[Buzz Bloom]

See this Wikipedia article (which is linked to from the one on the Sleeping Beauty problem that is linked to in the OP of this thread):

Hi Peter:

I get the relationship between "credence" and "probability", but I do not get any relationship to wagering from the puzzle statement. I do nopt see from the puzzle statement that Sleeping Beauty is to make a wager.

Regards,
Buzz

 

[PeterDonis]

I get the relationship between "credence" and "probability",

Did you read the Wikipedia article? It describes the relationship between credence and wagering.

I do nopt see from the puzzle statement that Sleeping Beauty is to make a wager.

The puzzle statement does use the term "credence", which, according to at least one common definition of that term, implies a wager, as described in the Wikipedia article on credence that I linked to. This has been discussed at great length in this thread, so I'm afraid you'll have to read the 500 posts if what's in the Wikipedia articles isn't enough.

 

[Buzz Bloom]

Did you read the Wikipedia article? It describes the relationship between credence and wagering.

The puzzle statement does use the term "credence", which, according to at least one common definition of that term, implies a wager

Hi Peter:

Yes I did, and the discussion there is about ways to think about solving the puzzle, not about anything inherent in the problem statement itself. As I interpret the discussion, it was that SB's credence could be determined by the limits she used to decide what bets she would decide are profitable. Thus (if she were a thirder) she would accept a bet paying her odds of 2:1 + a small amount more, but not 2:1 + a small amount less. The determination of a boundary between accept and not could be determined by the interview without any bets actually being made. Thus making a wager is not implied, but wagers might be discussed.

Regards,
Buzz

 

[PeterDonis]

SB's credence could be determined by the limits she used to decide what bets she would decide are profitable

Yes, and that is how the term "wager" is being used in this discussion. You can think of it as actual wagers, or as hypothetical ones; it doesn't matter. The question of whether 1/3 or 1/2 is the correct credence is the same either way.

 

[bahamagreen]

Do both halfers and thirders agree that she would answer...?

What is the chance that the coin landed heads?
"One half"

What is the chance that today is Monday?
"Two thirds"

What is the chance that the coin landed heads and that today is Monday?
"One third"

If betting is included as part of the experiment and SB's rationale, what is her credence Wednesday when she is debriefed and informed, "Oh, you didn't win any money; of course you can't recall anything about it but you were incorrect".

 

[Stephen Tashi]

I'm coming into this late, but it seems very clear to me the correct answer is 1/3, on the simple grounds that Sleeping Beauty knows that if she guesses "the coin landed heads" every time she is awakened, and if the experiment is repeated every week for a year, then she will clearly have been correct 1/3 of the time, and none of those events will seem any different to her. So that's 1/3 credence, and I can see no other meaningful way to define the concept of "credence."

1/3 is the correct answer to some questions - but which questions?

The answer 1/3 is a correct answer to the question: "In large number of repetitions of Sleeping Beauty experiment, what is the expected ratio of ( The number of situations that arose when Sleeping Beauty was awakened when the coin landed heads) to the (Total number of situations that arose in the experiment)?

The answer (1/3)$ is also correct answer to the question: When awakened, what is a fair price for Sleeping Beauty to pay for an agreement that she must always pay that price when awakened and only gets $1 on those awakenings where the coin landed heads.

The controversy (in some minds) is whether Sleeping Beauty should interpret the question "What is your credence that the coin landed heads" literally. If she takes a literal interpretation, she attempts to apply the definition of credence given by https://plato.stanford.edu/entries/probability-interpret/#SubPro

This boils down to the following analysis:

Your degree of belief in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.

She must define a price for the bet "you get $1 if the coin landed heads" , not for the agreement "you pay the price every time you are awakened and you get $1 if the coin landed heads.".

The assumption that Sleeping Beauty is rational compels her to pay the same price for a bet each time she awakens because she can't tell one awakening from another. Thus if she pays X for a bet on Monday and the coin landed tails, she will (i.e must) pay X for the same bet when it is offered on Tuesday. Sleeping Beauty knows the way the experiment is conducted so she knows of such a possibility.

What Sleeping Beauty needs is a bet on "The coin landed heads" that has a simple payoff of $1 instead of additional consequences.

 

[.Scott]

But there hasn't been a satisfactory account of why the probability of heads changes.

The numbers work out the same as in Sleeping Beauty. But in this case, the fact that I am picked is additional information that changes the conditional probability of heads. In the Sleeping Beauty case, the fact that she is asked the probability upon waking is no additional information, since it was a certainty that that would happen.

The problem is not that she has additional information upon being awakened, it is that she has less. Or perhaps it is better to explain it as having different information about the coin toss.

Before being put to sleep the first time, she knows it's 50/50. And she knows that she will be awakened once or twice. But when she is awakened, she doesn't know which day it is and she doesn't know if this is one of one or one of two.

Clearly, if she makes a wager each Monday or Tuesday when she is awoken, she should use the 33.3:66.6 odds, not the 50:50.

In general, there is a 50:50 shot that the flip of a coin will be heads or tails. But once you add information about the outcome of the coin toss, the odds change. Credence is determined by what you know and don't know. So as that changes, so does the credence.

It's the same for those conducting the experiment. At the start of the experiment, they would say there is a 50:50 shot that it was heads. But once they flip the coin, it changes to 100:0 one way or the other.