Undergrad 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.