I The Sleeping Beauty Problem: Any halfers here?

[Dale]

In fact, the problem doesn't exclude the real life possibility that interviews vary in duration, even within one run of the experiment.

I covered that possibility too.

Considering time points out that the Sleeping Beauty problem is ill-posed as probability problem. (It's status as a problem about "credence" may be a different story.)

Credence is a probability. It satisfies all of the axioms of probability. The sleeping beauty problem is well posed, the point of the consideration of time is to show that it doesn't matter. It drops out of the ratio.

If two things are not explicity given as equiprobable, the additional assumption that they are equiprobable needs to be stated by the problem solver.

Alternatively, a method may be used which does not depend on whether or not they are equiprobable!

 

[stevendaryl]

Perhaps among those whose discuss problems in "credence" there is a ground rule that says one is always permitted to apply The Principle Of Indifference. (And perhaps a discussion of "credence" is more on-topic in this thread that a discussion of mathematical probability, but at the moment, I am only discussing whether the Sleeping Beauty problem is well-posed as a problem in mathematical probability. )

The procedure---what actually is done--is completely well-specified, so I don't understand this business about selection procedures. There is no selection procedure, other than the initial coin toss. The question is whether is a non-arbitrary way for Sleeping Beauty to quantify her uncertainty about her situation when awakened. I would say that the principle of indifference is definitely a non-arbitrary means of solving the problem. If you want to say that it's an additional assumption, I guess I would concede that point. Although historically, probability theory was originally developed by exploiting the principle of indifference, and the more general notion of probability was developed to generalize to cases where the principle was too weak to give an answer.

 

[Stephen Tashi]

If no probability value is given in a mathematical probability problem, the PoI is the only way to supply one. For example, you are applying it when you say the probability that the coin will land with heads showing is 50%.

The fact that making an assumption is the only way to solve mathematical probability problem demonstrates that the problem is ill-posed.

One does not automatically assume a coin in a mathematical probability problem has an equal probability of landing heads or tails. It should be explicitly stated that it is a "fair" coin.

Kolmogorov didn't tell us how to assign values to probabilities, he just laid out the properties the values need to have.

Nevertheless, for a specific probability space to be explicitly described in problem, the details of assigning probabilities must be given.

 

[stevendaryl]

If no probability value is given in a mathematical probability problem, the PoI is the only way to supply one. For example, you are applying it when you say the probability that the coin will land with heads showing is 50%.

Kolmogorov didn't tell us how to assign values to probabilities, he just laid out the properties the values need to have.

I would have to agree with @Stephen Tashi that the POI is an additional assumption, although in most informal word problems involving probability, it's implicit that you're supposed to use it whenever applicable.

 

[Dale]

The fact that making an assumption is the only way to solve mathematical probability problem demonstrates that the problem is ill-posed

But making that assumption is not the only way to solve it.

 

[Stephen Tashi]

The procedure---what actually is done--is completely well-specified, so I don't understand this business about selection procedures.

I don't see any business about selection procedures in the passage you quoted.

But, yes, I speak in terms of selection procedure for selecting what situation exists when Sleeping Beauty is awakened. If there is a mathematical probability distribution for the situation when Sleeping Beauty is awakened then it describes how to stochastically "select" the situation from a probability distribution defined on situations. So the answer to the question "What is the probability that the coin landed heads when Sleeping Beauty is awakened" is approached by find a probability distribution that can be interpreted as defining how we stochastically "select" what situation exists when Sleeping Beauty awakes.

I would say that the principle of indifference is definitely a non-arbitrary means of solving the problem. If you want to say that it's an additional assumption, I guess I would concede that point. Although historically, probability theory was originally developed by exploiting the principle of indifference, and the more general notion of probability was developed to generalize to cases where the principle was too weak to give an answer.

This must be a difference in mathematical cultures. In the culture of my education, there is no axiom of probability theory that says the Principle Of Indifference can always be used on occasions where it applies - in the sense that lack-of-information-to-make-a-distinction between events can be used to assert equality of their probabilities. There can be other types of symmetry arguments among variables that allow their values to be deduced, but these arguments are applications of given information, not deductions from lack-of-information.

If it were demonstrated that solving a problem using the Principle of Indifference always gives the same answer to a problem, no matter how the solver formulates a solution, then I would say the problem has a unique solution when we allow the free use of the Principle of Indifference. However, I'm not aware that such a theorem has been proven - for a general class of problems or even for one specific problem.

If we invent 10 different problems equivalent to a given interpretation of the Sleeping Beauty problem and solve each of them by applying the Principle Of Indifference at various stages of the solution, and all 10 produce the same answer then this is empirical evidence suggesting that the Sleeping Beauty Problem has a unique solution when free use of the Principle Of Difference is allowed. However, I don't see the empirical result as a mathematical proof that the Sleeping Beauty Problem has a unique solution. How do we know there isn't an 11th problem that is equivalent to the Sleeping Beauty Problem and gives a different solution?

 

[Stephen Tashi]

The sleeping beauty problem is well posed, the point of the consideration of time is to show that it doesn't matter. It drops out of the ratio.

It only drops out the ratio if you have made assumptions about the ratio of the durations of interviews. You also must assume the probability of a situation is proportional to the duration of its associated interview.

 

[Dale]

You also must assume the probability of a situation is proportional to the duration of its associated interview.

I left that as an exercise for the interested reader, but it drops out in that case too.

 

[JeffJo]

The fact that making an assumption is the only way to solve mathematical probability problem demonstrates that the problem is ill-posed.

The fact that you would understand that a fair coin was intended, whether or not it was stated explicitly, demonstrates that any problem with how well it is posed is something you are creating yourself. Besides...

1) The best answer for "What is the probability that this coin, which I know to be unfair, will land Heads" is still 50%. Because the PoI can still be applied to its unknown bias, and
2) If conclusive implication is not enough for you, it should also be explicitly stated that Beauty cannot tell if it is Monday or Tuesday, which makes the application of the PoI to the day explicit.

 

[stevendaryl]

I don't see any business about selection procedures in the passage you quoted.

I'm disputing that there is any need for such a thing. The situation is completely specified: what happens to Sleeping Beauty, what she knows and doesn't know, what she's being asked. Talk about selection procedures beyond that is not relevant, except to the extent that you are allowed to solve the problem by converting it to an equivalent, more traditional probability problem in terms of selecting an event at random.

But, yes, I speak in terms of selection procedure for selecting what situation exists when Sleeping Beauty is awakened. If there is a mathematical probability distribution for the situation when Sleeping Beauty is awakened then it describes how to stochastically "select" the situation from a probability distribution defined on situations. So the answer to the question "What is the probability that the coin landed heads when Sleeping Beauty is awakened" is approached by find a probability distribution that can be interpreted as defining how we stochastically "select" what situation exists when Sleeping Beauty awakes.

That's fine, as long as you keep in mind that there is actually no selection going on.

This must be a difference in mathematical cultures. In the culture of my education, there is no axiom of probability theory that says the Principle Of Indifference can always be used on occasions where it applies - in the sense that lack-of-information-to-make-a-distinction between events can be used to assert equality of their probabilities. There can be other types of symmetry arguments among variables that allow their values to be deduced, but these arguments are applications of given information, not deductions from lack-of-information.

That's a difference between pure and applied mathematics. If you're trying to apply probability theory to a (real problem or made-up thought experiment such as this one), you have to make additional assumptions before any of it is applicable. If you want to apply probability theory to the chances of getting a straight flush in poker, you have to make assumptions based on symmetry.

If we invent 10 different problems equivalent to a given interpretation of the Sleeping Beauty problem and solve each of them by applying the Principle Of Indifference at various stages of the solution, and all 10 produce the same answer then this is empirical evidence suggesting that the Sleeping Beauty Problem has a unique solution when free use of the Principle Of Difference is allowed. However, I don't see the empirical result as a mathematical proof that the Sleeping Beauty Problem has a unique solution. How do we know there isn't an 11th problem that is equivalent to the Sleeping Beauty Problem and gives a different solution?

Then the problem will be reopened. That doesn't bother me. Knowledge that isn't tautological is subject to revision.

 

[JeffJo]

In the culture of my education, there is no axiom of probability theory that says the Principle Of Indifference can always be used on occasions where it applies...

There is also no axiom of arithmetic that tells me how the operation of subtraction can be used when I want to weigh my cat by weighing myself, and then weighing myself holding the cat. These are applications of an abstract field of mathematics to a real-world problem, and so require an interpretation of the abstract concepts.

There is no axiom of probability theory that tells us how to assign values to probability. Period. All we have are these three guidelines:

1. Every probability is a non-negative number.
2. If A and B are disjoint events, the probability of the union of A and B is the sum of their individual probabilities.
   
3. The probability of the universal event is 1.

Everything else is outside the purview of probability theory. If you want to assign values of 1% and 99% to two indistinguishable events, probability theory itself won't stop you.

But in application, the interpretation of probability is likelihood. In application, if you want to assign unequal of values to two events, you have to justify that there is a difference between them. This process may be called "The Principle of Indifference," but all it really is, is the only way we have attach numbers to the abstract concept of "probability" when we deal with real-world problems. It is not a proven method in the abstract field, it is a name we apply to the recognition of a simple fact: that a difference is impossible in application.

And there are two forms: The weak, where you can't identify what the difference in likelihood is, even though the situations are dissimilar; and the strong, where you prove that the situations are the same in every way that can matter to likelihood. I haven't followed your discussions with others, about how to (or if necessary to) accomplish this proof with the ill-formed events that you think can carry over from the future of Sunday, to the present of a waking day. I have re-cast the problem into one that has to have the same answer, and where I can accomplish this proof. And you are using the perfectly valid objections to the weak PoI to ignore any application of the strong PoI.

... in the sense that lack-of-information-to-make-a-distinction between events can be used to assert equality of their probabilities.

That's the weak PoI. I use the strong PoI.

There can be other types of symmetry arguments among variables that allow their values to be deduced, but these arguments are applications of given information, not deductions from lack-of-information.

And the difference between the weak, and strong, is that the stong demonstrates that the given information is equivalent.

How do we know there isn't an 11th problem that is equivalent to the Sleeping Beauty Problem and gives a different solution?

Because we have proven, in the 10, that there is an equivalence. If this hypothetical 11th exists - and you haven't found one - then what we have is a true paradox, not an ambiguous problem.

 

[Stephen Tashi]

That's fine, as long as you keep in mind that there is actually no selection going on.

I'm not familiar with with any technical terminology implied by term "selection". Are you using one?

Nothing in the Sleeping Beauty problem is actually going on. In the spirit of applied mathematics, we can seek a probability model for an answer to the question posed and implement it as a simulation where a particular situation is "realized", "selected" - or whatever - from some probability distribution. From that model, P(H|A) can be determined.

I understand that there can be an abstract, philosophical view where people seek to determine P(H|A) without determining P(the situation | A).

That's a difference between pure and applied mathematics. If you're trying to apply probability theory to a (real problem or made-up thought experiment such as this one), you have to make additional assumptions before any of it is applicable. If you want to apply probability theory to the chances of getting a straight flush in poker, you have to make assumptions based on symmetry.

I agree that solving real life problems involves making assumptions, but the precise statement of a problem as a mathematical problem describes what is assumed. In problems of poker hands, one should state the assumption that all deals have equal probability. I have yet to see any supposed solver of the Sleeping Beauty in this thread state the problem as a well posed mathematical problem. (Such a statement would include all the assumptions that are being made). Instead, people (including "thirders") begin discussing the problem as if it were well posed then purport to deduce an answer only from things explicitly stated in the problem.

There is disagreement on the "ground rules" for this discussion. In the first place, some posters (like myself) take a non-philosophical view and concentrate on computing a probability instead of a "credence". Some posters (of both the philosophical and non-philosophical bent) take the viewpoint that whenever they encounter a symmetry in the form of equal-lack-of-information, they may apply the Principle of Indifference. If one is going to take that view, the problem they are solving can be clearly presented as a mathematical problem by stating what is assumed at the outset instead of launching into a proof of a particular answer and then, in the middle of it, saying "Here I will apply the Principle of Indifference", thus adding an additional assumption in the middle of a purported proof.

Then the problem will be reopened. That doesn't bother me. Knowledge that isn't tautological is subject to revision.

That may be a philosophical outlook, but it is not the viewpoint of mathematics. A hundred examples of the truth of Fermat's Last Theorem were not taken as proof of the theorem. The problem of proving it wasn't closed.

 

[Stephen Tashi]

There is no axiom of probability theory that tells us how to assign values to probability. Period.

All we have are these three guidelines:

1. Every probability is a non-negative number.
2. If A and B are disjoint events, the probability of the union of A and B is the sum of their individual probabilities.
   
3. The probability of the universal event is 1.

There is no axiom of the real numbers that tells us how to assign values to variables. However, mathematical problems of solving for variables are stated by giving additional information about the variables (e.g. systems of simultaneous equations). Problems in probability are not an exception to this policy. It is not correct to say that the only information in a well-posed probability problem is that variables representing probabilities obey the axioms for a probability space.

But in application, the interpretation of probability is likelihood. In application, if you want to assign unequal of values to two events, you have to justify that there is a difference between them.

You can take that philosophical view and perhaps people who discuss "credence" have such ground rules.

The ground rules in mathematics (including probability theory) are different. Suppose we have a problem with variables X,Y,Z and there is no statement that X and Y have different values. A person claims that the unique solution to the problem is X = 2, Y = 2, Z = 5. Another person shows that X = 1, Y = 2, Z = 3 is a solution that makes all the given information in the problem true. The fact that the solution X =1 , Y = 2, Z = 3 violates an un-given consideration that X and Y should have the same value is irrelevant. The solution to the problem is proven not unique.

Applying the latter outlook to the Sleeping Beauty Problem, a person who claims the "halfer" solution is wrong must demonstrate what explicit statements in the problem the "halfer" solution violates. The fact that the "halfer" solution makes two variables unequal that were not stated to be unequal does not show the "halfer" solution is wrong. The proper way to show the "halfer" solution is the wrong answer to an interpretation of the Sleeping Beauty Problem is explicitly state which variables are assumed to be equal in that interpretation - and this should be done in the statement of the problem, not halfway through some attempt to solve it!

Saying the "halfer" is wrong is a different consideration that saying the "halfer" solution is not unique or that the solution is obtained by making unwarranted assumptions. The correctness of a solution is evaluated only by using the values proposed by the solution, not how those values were derived. We can evaluate the correctness of a solution that is just somebody's guess. When we evaluate whether the procedure use to obtain the "halfer" solution is a rigorous proof of its result, we can object to unwarranted assumptions. But showing someone's procedure for arriving at a solution is not correct reasoning does not prove the solution that was reached is wrong.

And there are two forms: The weak, where you can't identify what the difference in likelihood is, even though the situations are dissimilar; and the strong, where you prove that the situations are the same in every way that can matter to likelihood.

That's an important distinction and I'm glad you explained it. We need to be clear on what it takes to prove situations are the same. In a mathematical problem symmetries are most often stated as invariances. For example, we might be given that F(x,y) = F(y,x). In a long verbal discussion about a probability problem that lacks precise mathematical statements, it isn't clear to me what criteria the strong PoI sets for showing two events are "the same in every way". (If they were the same in every way, we'd be talking about one event, not two events.)

I have re-cast the problem into one that has to have the same answer, and where I can accomplish this proof. And you are using the perfectly valid objections to the weak PoI to ignore any application of the strong PoI.

That's the weak PoI. I use the strong PoI.

Another glitch in this thread is that no mathematical definition has been stated for what it means for two probability problems to be "equivalent". I can understand your statements by using imprecise "common language" interpretations of "re-cast" and "strong PoI" However, what they mean in terms of mathematics needs clarification.

Because we have proven, in the 10, that there is an equivalence.

The ten examples I mentioned were not proofs of an equivalence. There were examples of first creating a new problem that is definitely equivalent (once that concept is defined!) to the original problem. Then the ten examples solve the new problem by using the Principle of Indifference wherever it is applicable. The important variation among the ten examples is that the Principle of Indifference is applied on different occasions.

I am not defining "equivalent" to mean "has the same numerical answer". We need to work on the definition of "equivalent", but the general concept is that two problems which use different probability spaces are equivalent whenever we can define the events in one probability space in terms of events in the other. The fact that two problems are equivalent should imply that the the answer produced by one can be matched to an event with the same numerical probability in the other. However, having the same numerical answer is not the definition of being equivalent.

The ten examples are not proofs that the solution to the problem is unique. For example, suppose that instead of the ground rule that one may apply the PoI whenever it is possible, we adopt the crazy ground rule: "Whenever a variable begins with a capital letter its value can be set to 13." Finding ten examples of solving an equivalent problem by using different variable names and getting the same answer is not a proof that using such a ground rule always produces the same answer to the original problem.

 

[JeffJo]

There is no axiom of the real numbers that tells us how to assign values to variables. However, mathematical problems of solving for variables are stated by giving additional information about the variables (e.g. systems of simultaneous equations).

You do understand that you don't "solve for" the value of a random variable, don't you? So "problems in probability" certainly are an exception to whatever policy you think you are applying to them.

Suppose we have a problem with variables X,Y,Z and there is no statement that X and Y have different values.

No.

Suppose we have probability constants Pr(A) and Pr(B), proof that they can't be different, and the axiom that Pr(A)+Pr(B)=1. Can you, or can you not, deduce what Pr(A) and Pr(B) are? This is the strong PoI.

a person who claims the "halfer" solution is wrong must demonstrate what explicit statements in the problem the "halfer" solution violates.

First, the halfer must define what probability space he thinks that solution applies to, and why it describes a woman who doesn't know if it is Monday or Tuesday.

But you are using this argument to evade the fact that I have provided equivalent problems, with defendible solutions.

The fact that the "halfer" solution makes two variables unequal that were not stated to be unequal does not show the "halfer" solution is wrong.

The fact that, if you tell the halfer that it is Monday, that halfer must update her solution to Pr(H|Monday)=2/3 does show it.

The proper way to show the "halfer" solution is the wrong answer to an interpretation of the Sleeping Beauty Problem is explicitly state which variables are assumed

to be equal in that interpretation

The prior probabilities, when the lab tech showed up to work this morning but before determning whether Beauty should be wakened, for whether today was Monday or Tuesday, were equal. The prior probabilities at that time, for wheter the coin was Heads or Tails, were also equal. The two random variables are independent, so the prior probability for each combination is 1/4. The information "Gee, I was wakened" eliminates (H,Tue), so the updated probability for heads is 1/3.

The halfer's mistake is one of interpretation, not probability theory. It treats "Monday will happen" and "Tuesday will happen," evaluated on Sunday Night, as random variables. They are not, they are certainties.

The halfer is wrong.

That's [strong vs. weak PoI] an important distinction and I'm glad you explained it.

It's about the fourth time I have.

We need to be clear on what it takes to prove situations are the same.

They occur under symmetric situations. The situation for each of my four Beauties differ only in symmetric ways. Each is in an equivalent of the OP. One of the three who is awake can answer "yes" to whether the event that is happening at the moment is the one she was asked to provide a credence for.

What's the problem?

In a mathematical problem symmetries are most often stated as invariances.

But don't have to be. But, if you want, the OP is invariant when you swap Monday for Tuesday. The OP is invariant when you swap Heads for Tails. Thus the four Beauties are in invariant situations, and one is in the OP itself. One of the three who is awake can answer "yes" to whether the event that is happening at the moment is the one she was asked to provide a credence for.

For example, we might be given that F(x,y) = F(y,x). In a long verbal discussion about a probability problem that lacks precise mathematical statements, ...

No, its one where you refuse to recognize what the precise mathematical situations need to be.

The ten examples I mentioned were not proofs of an equivalence.

The example I assume you implied were ones that were invaraiant under the changes made.

I am not defining "equivalent" to mean "has the same numerical answer". We need to work on the definition of "equivalent", but the general concept is that two problems which use different probability spaces are equivalent whenever we can define the events in one probability space in terms of events in the other.

And I have done that.

The ten examples are not proofs that the solution to the problem is unique.

If a solution is correct, it is either unique or probability theory is inconsistent.

+++++

Let's try this again. Four women participate in an OP-like experiment. They use the same coin, and 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); and they are asked about the coin result in that set of circumstances, not necessarily "Heads".

Question for you: assuming they are all perfectly logical, should each arrive at the same conclusion about the solution to the problem? I want a "yes" or a "no," not equivocation about whether it is ambiguous or not, or what that answer is.

 

[Dale]

Applying the latter outlook to the Sleeping Beauty Problem, a person who claims the "halfer" solution is wrong must demonstrate what explicit statements in the problem the "halfer" solution violates.

The halfer solution violates the statement in the problem that Beauty is rational.

 

[Stephen Tashi]

The halfer solution violates the statement in the problem that Beauty is rational.

Why? - because Sleeping Beauty doesn't think about distributions of times and time intervals?

 

[Stephen Tashi]

You do understand that you don't "solve for" the value of a random variable, don't you?

A random variable is specified by a probability distribution on a set of events. We do solve for the values of those probabilities.

Suppose we have probability constants Pr(A) and Pr(B), proof that they can't be different, and the axiom that Pr(A)+Pr(B)=1. Can you, or can you not, deduce what Pr(A) and Pr(B) are? This is the strong PoI.

If that is "strong PoI", it is ordinary reasoning provided there is mathematical proof that the two numbers must have the same value. It is not a mathematical proof to say “No information is given to distinguish between Pr(A) and Pr(B), therefore they must have the same value.

First, the halfer must define what probability space he thinks that solution applies to, and why it describes a woman who doesn't know if it is Monday or Tuesday.

As far as describing probability spaces goes, let he who is without sin...
Each halfer will have to speak for himself, but here is one formulation of the problem.
The following events are defined:
H = the coin landed heads

T = ~H = the coin landed tails

(using "~" to indicate the complement of a set)
Wm = SB is awakened on Monday

Wt = SB is awakened on Tuesday
a = today SB is awakened

m = today is Monday

t = ~m = today is Tuesday

(i.e. "today is Tuesday" is the same event (in the context of the problem) as the event "today is not Monday".
The following probability spaces are defined in terms of those events.
The coin space is defined on the mutually exclusive events H,T

P(H) = 1/2

P(T) = 1/2
The conduct of the experiment has the probability space defined on the mutually exclusive events Eh, Et:.

Eh = (H and Wm and ~Wt) , P(Eh) = 1/2

Et = (T and Wm and Wt) , P(Et) = 1/2
The probability space required by question is defined on the mutually exclusive events ("situations") s1,s2,...s8 . One possible "halfer" space is:

s1 = ( H and m and a), P(s1) = 1/2

s2 = ( H and m and ~a), P(s2)= 0

s3 = ( H and t and a), P(s3) = 0

s4 = ( H and t and ~a), P(s4) = 0

s5 = ( T and m and a), P(s5) = 1/4

s6 = ( T and m and ~a), P(s6) = 0

s7 = ( T and t and a), P(s7) = 1/4

s8 = (T and t and ~a), P(s8) = 0.
The above probability space can be implemented by a stochastic process for selecting what happens “today”. To repeat the description for an earlier post: Flip the coin to determine which situations arise in the experiment. From those situations where Sleeping Beauty is awake, pick one of the situations at random, giving each the same probability of being selected.
Notes:

There is no information given in the Sleeping Beauty problem that gives the value for P(s2) and, in particular, there is no information that says it cannot be zero. If a person wishes to imagine some stochastic process that determines what day of the week “today” is when Sleeping Beauty is asleep , they are welcome to do so, but the problem does not say such a process has a particular distribution. So a “halfer” is free to assign P(s2) = 0. (This is not a claim that the “halfer” obtains the unique solution to the problem. He obtains one of several solutions.)

The problem specifies no relation between the events Wm, Wt and the events m,a.

The proposed "halfer" answer is P( H | a) = P( H | ( s1 or s3 or s5 or s7 ) )= 1/2.
If you wish to introduce a probability space involving what Sleeping Beauty knows, you'll have to explain how to do that. The problem states Sleeping Beauty definitely knows certain things when she awakes and doesn't know others. There is nothing stochastic about what she knows. Each time she is awakened she knows the same things.---

The fact that, if you tell the halfer that it is Monday, that halfer must update her solution to Pr(H|Monday)=2/3 does show it.

Are you asking for an update of P(H|a) to P(H| (a and m))?P(H| (a and m)) = P(H | s1) = 1

Edit: No, I'm wrong.
P(H | (a and M)) = P(H| (s1 or s5)) =2/3
However, there is nothing wrong about that.

 

[Dale]

Why? - because Sleeping Beauty doesn't think about distributions of times and time intervals?

Because if she believes the probability of 1/2 then she would either have to bet contrary to her beliefs or expect to lose money.

 

[Stephen Tashi]

Because if she believes the probability of 1/2 then she would either have to bet contrary to her beliefs or expect to lose money.

I agree, if she is required to make two even-odds bets on heads when the coin lands tails. However, no wagering scheme is given in the problem.

If a fair coin is tossed, and I lose 2 dollars if it lands heads and 1 dollar if it lands tails, I would always guess tails. There is nothing in that sort of wager that involves being awakened, amnesia etc. Guessing tails is "rational" but it doesn't imply the probability of tails is greater than the probabilitiy of heads.

 

[Dale]

Guessing tails is "rational"

And Beauty is rational, per the problem description.

As you said above, all you need to do is to plug the answer back in and see if it is a solution. In this case, 1/2 is not. The halfer solution is inconsistent with the given information.

If a fair coin is tossed, and I lose 2 dollars if it lands heads and 1 dollar if it lands tails, I would always guess tails. ... Guessing tails is "rational" but it doesn't imply the probability of tails is greater than the probabilitiy of heads.

Btw, the reason this is rational is precisely because your credence for heads is higher than 2:1 against. So here you are, in fact, recognizing the direct link between rational credence and acceptable betting odds.

 

[Stephen Tashi]

The fact that, if you tell the halfer that it is Monday, that halfer must update her solution to Pr(H|Monday)=2/3 does show it.

The objection to the update P(H| (a and m)) = 2/3 seems to be the opinion that when Sleeping Beauty is awakened on a Monday, the actual probability that the coin landed heads is 1/2. However that is not true if the "halfer" method of picking the situation "today" is used. In the "halfer" model for what happens "today", the probabilities are as stated. (In the halfer model , when coin lands heads, "today" must be Monday because it's the only day in the experiment when Sleeping Beauty is awakened. If the coin lands tails, "today" might be Monday or Tuesday.)

The Sleeping Beauty problem does not give the information " Whenever Sleeping is awakened and it's Monday the probability the coin landed heads is 1/2".

If it is asserted that P(H | (a and m)) = 1/2 can deduced without adding assumptions to the information given in the problem ( including adding them by using the weak Principle of Induction) then it should be possible to show P(H| (a and m) = 1/2 in straightforward manner using standard mathematical reasoning.

 

[Stephen Tashi]

And Beauty is rational, per the problem description.

That consideration is relevant to determining "credence". However it is not needed if there is a unique answer for P(H|a) that can be computed by applying probability theory only to the information given in the problem. We who attempt to solve for P(H|a) know exactly what Sleeping Beauty does when she is awakened. If someone can deduce the value of P(H|a) from the given information, he doesn't need the assumption "I am rational".

As you said above, all you need to do is to plug the answer back in and see if it is a solution. In this case, 1/2 is not. The halfer solution is inconsistent with the given information.

No. There is no betting scheme given in the problem.

Btw, the reason this is rational is precisely because your credence for heads is higher than 2:1 against. So here you are, in fact, recognizing the direct link between rational credence and acceptable betting odds.

- And the distinction between "credence" in a probabilty and actual value of that probability.

I'm willing to discuss Sleeping Beauty's credence. (First, it would help to have a clear definition of "credence"). However, all I have asserted so far in this thread is that there no unique solution for the probability P(H | a) that follows from applying mathematical reasoning to the given information about probabilities of events. As I understand your position, you think there is a unique solution for P(H | a) if we treat the problem as problem involving physics - i.e. include considerations of time intervals amd selecting random times. While I don't follow your reasoning, I agree that considering a model of the physical situation implied by the problem could add to the "given" information about events.

 

[Dale]

That consideration is relevant to determining "credence"

Which is also part of the problem statement!

In order to get the halfer answer you are ignoring both the part of the problem statement where Beauty is given to be rational and the part where she is asked to state her credence!

You may object to the additional assumption of the PoI, but at least the thirders are not violating the given information.

No. There is no betting scheme given in the problem

It is part of the meaning of the word credence. See the links that I gave Peter Donis. The link between betting and credence is well established.

As I understand your position, you think there is a unique solution for P(H | a) if we treat the problem as problem involving physics - i.e. include considerations of time intervals amd selecting random times.

That is not my position at all, as I was exceptionally explicit to point out.

 

[Stephen Tashi]

In order to get the halfer answer you are ignoring both the part of the problem statement where Beauty is given to be rational and the part where she is asked to state her credence!

Guilty! - because the first thing to settle is whether there is an objectively correct value for P(H|a). As I understand it, several posters (including yourself) claim that P(H|a) can be computed from the information given in the problem without additional assumptions and without resorting to making more assumptions by using the weak version of the The Principle of Indifference.

You may object to the additional assumption of the PoI, but at least the thirders are not violating the given information.

I agree that the "thirder" solution is a solution. So is the "halfer" solution.

As mentioned in another post, the correctness of a solution is independent of how it is derived since we can check whether it satisfies the constraints that define a solution. A solution can be somebody's guess.

It is part of the meaning of the word credence. See the links that I gave Peter Donis. The link between betting and credence is well established.

I see long discussions of concepts in those links. I don't see any precise mathematical definition of "credence". Will someone please give a precise definition of "credence"? - or will attempts to be precise stir up even more controversy?

That is not my position at all, as I was exceptionally explicit to point out.

Ok, I can't figure out what your position is. Perhaps someone besides you can explain your position to me.

 

[Dale]

So is the "halfer" solution.

As mentioned in another post, the correctness of a solution is independent of how it is derived since we can check whether it satisfies the constraints that define a solution.

No it isn't. Plug it back in, it would not lead to rational betting behavior by Beauty, so it isn't a rational credence. It therefore does not satisfy the constraints.

I see long discussions of concepts in those links. I don't see any precise mathematical definition of "credence".

The long discussions are useful since they help understand what the community means when it uses the word "credence". The second link has the precise definition you are looking for.

Ok, I can't figure out what your position is

See post 255.

 

[stevendaryl]

I'm not familiar with with any technical terminology implied by term "selection". Are you using one?

I thought you were the one demanding that we specify a selection procedure. If you don't mean anything by those words, then I can't understand what your point was.

I agree that solving real life problems involves making assumptions, but the precise statement of a problem as a mathematical problem describes what is assumed. In problems of poker hands, one should state the assumption that all deals have equal probability. I have yet to see any supposed solver of the Sleeping Beauty in this thread state the problem as a well posed mathematical problem. (Such a statement would include all the assumptions that are being made). Instead, people (including "thirders") begin discussing the problem as if it were well posed then purport to deduce an answer only from things explicitly stated in the problem.

Turning the problem into a well-posed problem IS the problem to be solved. You're basically saying: You can't solve this problem because it doesn't give you the answer.

That may be a philosophical outlook, but it is not the viewpoint of mathematics.

This wasn't a problem of mathematics. The challenge is to turn it into one. Once again, you're complaining that the problem statement doesn't provide the solution.

 

[stevendaryl]

Guilty! - because the first thing to settle is whether there is an objectively correct value for P(H|a). As I understand it, several posters (including yourself) claim that P(H|a) can be computed from the information given in the problem without additional assumptions and without resorting to making more assumptions by using the weak version of the The Principle of Indifference.

The way I view it is that the challenge of such a problem is to translate it from an informal puzzle to something that can be approached mathematically. Which means coming up with plausible, robust rules for subjective probability. Being plausible and robust to me means that whatever principles you invoke in solving the problem must give sensible answers to related variants.

It isn't a mathematical problem, it's a philosophical problem. A mathematical problem starts off with: We have such-and-such objects and operations and relations obeying such and such rules. Derive this. Prove that.

But there is a second part of science or mathematics, which is coming up with the objects, operations, relations and rules to begin with, and justifying them. In the Sleeping Beauty problem, that's the hard part. The actual mathematics of computing conditional probabilities is pretty trivial, at least for as simple a situation as SB.

You're complaining that the problem hasn't been properly reduced to a mathematical statement that you can solve by plugging into known formulas. I'm telling you that the challenge is the first step: reducing it to a mathematical statement (and justifying why that reduction is sensible). That part is not mathematics. The second part, which is coming up with a number after the situation has been converted into a pure mathematical problem, is completely trivial.

 

[Stephen Tashi]

The second link has the precise definition you are looking for.

From second link of post #384 on page 30 of the thread:
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.

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

For example suppose the bet is that each time the owner of the bet is awakened in the experiment, the owner is paid (1/2)$ if the coin lands tails. This satisfies the defiinition of a bet that pays $1 if coin lands tails since the owner is awakened twice. As soon as she knows the conditions of the experiment Sleeping Beauty can evaluate the worth of the bet as (1/2)$ by using the knowledge that the probability of the coin landing tails is 1/2.

So her "credence" is 1/2, which is obtained by a calculation that assigns a probability of 1/2 that the coin lands tails.

However, in this bet, the "event E" that is referred to in the above definition, is not simply how the coin lands, but the also a bet that the subsequent chain of events happens. So the bet is: When awakened, I will be paid (1/2)$ if the coin lands tails and I will be awakened two times if the coin lands tails. If Sleeping Beauty is not informed that the event E also involves the occurrence "I will be awakened two times if the coin lands tails" then it isn't clear how to determine her credence..

Suppose the bet is that each time the owner of the bet is awakened during the experiment the owner is paid $1 when coin lands heads. Sleeping Beauty can evaluate the worth of that bet as (1/2)$ and the event E is "When awakened, I will be paid $1 if the coin lands heads and I will be awakened once". So the event E in this bet is also not purely a bet on the event that the coin lands heads but also a bet on the event that the owner of the bet will be awakened once.

A person could argue that a bet on whether the coin lands tails should be a bet only on the event E = "the coin lands tails", not on the event "the coin lands tails and you will be awakened twice". How can such a bet be formulated in the context of the Sleeping Beauty Problem? Sleeping Beauty is informed of the conditions of the experiment each time she is awakened, so a "rational" Sleeping Beauty does not forget the connection between the coin toss and the number of awakenings.

As a generality, if the event K always happens when event C happens can we evaluate a persons credence in the event C|X as being distinct from their credence in the event C| (K and X)?

An interesting aspect of the scenario is that a "rational" Sleeping Beauty can figure out the worth of the bet without participating in the experiment. So the event W = "Sleeping Beauty is awakened during one day of experiment" doesn't change her credence about either of the two above bets.. There is no change in her credence due to the event W.

 

[Stephen Tashi]

The way I view it is that the challenge of such a problem is to translate it from an informal puzzle to something that can be approached mathematically. Which means coming up with plausible, robust rules for subjective probability. Being plausible and robust to me means that whatever principles you invoke in solving the problem must give sensible answers to related variants.

I am essentially in agreement with that outlook. I see it as 3 activities, each of which has some subjective aspects.

1) Establish the objective information about probabilities given in the problem. Determine if that information sufficient to deduce P(H|A) then we're done. (or rather I'm done, because if "credence" doesn't agree with objective probability, I'm not interested in investigating the defects of credence).

To me, the most interesting aspect of this task to argue whether or not the SB problem is equivalent to a simple balls-in-urns problem. It "almost" is objectively equivalent to choosing from urns H and T at random then choosing a ball at random from the chosen urn. The question is whether the condition "SB is awakened" matches to the event "One ball is drawn from the selected urn at random and its color is amber" or does it describe the less specific event "All the balls are drawn from the selected urn and an observer of one of the draws sees that the chosen ball is amber" - or perhaps some other event.

2) If the objective information is insufficient to deduce P(H|a) then investigate whether P(H|A) can be computed by assuming the Principle of Indifference (by which I mean the version of the PoI that assumes "indistinguishable" events are assigned equal probabilities).

There is active controversy among experts about whether solving problems in this manner produces a unique solution. Experts argue about whether the PoI can lead to different results when applied in different ways to the same problem - this is discussed by people who debate whether paradoxes involving the PoI exist.

In both tasks 1) and 2) have been approached by proposing problems "equivalent" to the SB problem and solving the equivalent problem. However the validity of this technique has not been established. In the first place, we ( or I) don't know the mathematical requirements for two problems to be "equivalent". They haven't been stated. People simply propose an problem and declare it's "equivalent" to the SB problem. ( I don't dismiss the technique of transforming problems. I'm working on how to state the sample spaces of the problem you summarized in post #449. Normally, I wouldn't tackle something like that, but we're having triple digit temperatures here in southern NM, the house is at 80F in the afternoon, making it attractive to sit quietly in front of a computer.)

3) Solve the problem of determining "credence" that is actually posed by the SB problem.

Perhaps this can't be done until task 2) is accomplished. Perhaps this task is impossible if we can't formulate a "pure" bet for the event H|A.

Naturally, posters often don't declare which task they are investigating, so it hard to figure out what ground rules they are using when they offer solutions. An internet thread won't carry out such a vast research programme anyway, so I can't complain about the usual chaos.

 

[JeffJo]

A random variable is specified by a probability distribution on a set of events. We do solve for the values of those probabilities.

A random variable is an property that has an observable value (not necessarily a number) in every possible outcome, that can differ from instance to instance. We define outcomes by specifying the value of every random variable we choose to use, and events by sets of these values. A probability distribution is linked to the set of events, not the random variables themselves.

But I will paraphrase what started this diversion: There is no axiom of probability that tells us how to solve for probabilities. There are axioms that tell us three properties we can include in that solution, but by themselves they do not allow for a solution. We either have to be told to assume probabilities, or find a way to apply the PoI.

If that is "strong PoI", it is ordinary reasoning provided there is mathematical proof that the two numbers must have the same value.

And unless you think a proof that the values can't be different is not proof that they must be the same, this is what the strong PoI is. It is not an axiom or an assumption, it is just "ordinary reasoning" (your words) once you have proven the values can't be different.

It is not a mathematical proof to say “No information is given to distinguish between Pr(A) and Pr(B), therefore they must have the same value.

That's the weak PoI. The strong is when you show that the factors that allow variation are the same.

... here is one formulation of the [OP].

The following events are defined:
H = the coin landed heads
T = ~H = the coin landed tails
(using "~" to indicate the complement of a set)
Wm = SB is awakened on Monday
Wt = SB is awakened on Tuesday.

Wm and Wt are ambiguous events since they cannot be evaluated in the posterior. And that's what I said was wrong with every halfer argument. They are only definable in a prior that considers both days to be included in experiment, which is not the correct prior to use. In that prior,they need to be phrased like "SB was/will be wakened on Monday", which isn't a random occurrence; and "SB will be wakened on Tuesday," which is the same event as T, so your set operations are probably faulty.

a = today SB is awakened

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.

m = today is Monday
t = ~m = today is Tuesday

And the reason your Wm and Wt are invalid, is that if you make these be events in the prior, they can only be evaluated on a single day. But your Wt applies to the set of two days, since it implies Wm. It isn't the OP that is ambiguous, it is how you define events for it. And this makes your attempt to define probability spaces from these events irrelevant.

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. There is a trivial set of events that are well-defined in this present-tense prior. I'm going to use the notation I'm familiar with: where outcomes and events are specified by indicating value for random variables. You called one such random variable a "coin space." The other is ill-defined in your system, which is probably why you didn't call it a "day space," and didn't carry Wm and Wt through to your probability spaces.

Random Variables: COIN, TODAY.

COIN has a value in the set {H,T}. The statement that the coin is fair only means that the factors that could lead to either outcome are the same. It is the PoI that says, once this is given, that their prior probabilities are 1/2.

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.

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

Being awake is an observation. As an event, it is only useful to define the condition. Its conditional probability, not its prior probability as claimed by halfers, is 1. Its prior probability is the sum of the prior probabilities of the outcomes that contribute to it: {(H,Mon),(T,Mon),(T,Tue)}, or 3/4. Because it is the condition SB has observed, she can update each of those [prior probabilities by dividing by 3/4. This makes the posterior probability of each remaining, possible combination 1/3.

One possible "halfer" space is [edited for space]:
s1 = ( H and m and a), P(s1) = 1/2
s4 = ( H and t and ~a), P(s4) = 0
s5 = ( T and m and a), P(s5) = 1/4
s7 = ( T and t and a), P(s7) = 1/4

Note that you intend these to be posterior probabilities, sicne you said Pr(s4)=0. Note also that you got these values by applying the PoI separately to {(H and m and a), (T and (m or t) and a)}, and then to {(T and m and a), (T and t and a)}.

The problem is, the PoI has to be applied in the prior, not the posterior. Pr(s4) is not zero in the prior. So even if I accepted these as a valid and consistent set of events (I don't), the application is wrong.

If you wish to introduce a probability space involving what Sleeping Beauty knows, you'll have to explain how to do that.

I have. Several times.

I've also created an equivalent problem, that's skirts all these issues.

Edit: No, I'm wrong.
P(H | (a and M)) = P(H| (s1 or s5)) =2/3
However, there is nothing wrong about that.

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?

+++++

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.

 

[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?