I The Sleeping Beauty Problem: Any halfers here?

[Dale]

Just the prior of heads doesn't answer the question, though. We can write:

P(heads|awake) = P(heads | awake & monday) P(monday | awake) + P(heads | awake & tuesday) P(tuesday | awake)

The second term is zero (since it's impossible for it to be heads if Sleeping Beauty is awake on a Tuesday). So we have:

P(heads | awake) = P(heads | awake & monday) P(monday | awake)

At this point, I would say that P(heads | awake & monday) = P(heads). Knowing that you are awake and that it is Monday doesn't tell you anything about whether the coin is heads or tails. So we have finally:

P(heads | awake) = P(heads) P(monday | awake) = 1/2 P(monday | awake)

So the two conditionals I think are closely related.

I do agree that they are closely related, but they are not the same. One of the problems in this thread is the wide variety of alternative scenarios proposed, which seem to be having the opposite effect as intended regarding clarifying the original scenario. So I suggest sticking to the scenario as specified in the Wikipedia article. There Beauty is asked her credence about the coin being heads. So the clear appropriate prior would be the 0.5 prior probability that a fair coin toss is heads. The prior probability to which day it is is not as clear and not necessary for the problem.

 

[Boing3000]

I gave you the formula:

P(heads | awake) = P(heads | awake & Monday) P(Monday | awake) + P(heads | awake & Tuesday) P(Tuesday | awake)

That formula is a theorem of conditional probability (together with the assumption that it's either Monday or Tuesday). What else do I need to explain about it?

You could simple have answered the question "explain your statement that she can test her amnesia or "update her information", instead of running in circle a get back to this formula which does not apply to the Sleeping Beauty problem.
We are not asked about probability of her being awake any day. She is ask only when she is awake, some questions about a coin flipped once. Why bother if her answer may not change from day to day ?

You would not even have taken the time to read the problem, if Peter hasn't forced you to (thanks to his credence). Read the problem, and try to understand it, your formula does not apply to it, because the problem is NOT defined enough for that (also read post#67)

If P(heads | Monday & awake) = 1/2 and P(Monday | awake) < 1, then it follows that P(heads | awake) < 1/2

And again, P(Monday | awake) = 1, because of the drug.

 

[PeterDonis]

Beauty is asked her credence about the coin being heads. So the clear appropriate prior would be the 0.5 prior probability that a fair coin toss is heads. The prior probability to which day it is is not as clear and not necessary for the problem.

I'm not sure I understand. If the only prior that is relevant is P(Heads) = 1/2, and no probabilities related to which day it is are relevant, then you are basically agreeing with the halfer argument. Is that your intent? Or are you just saying that P(Monday|Awake) = 2/3, which is the crucial factor needed for the thirder's argument, is not a "prior" but something else?

 

[PeterDonis]

P(Monday | awake) = 1, because of the drug

This is not correct. P(Monday|Awake) is the conditional probability that it is Monday, given that Beauty is awake. It is not the conditional probability that Beauty is awake, given that it is Monday; that would be P(Awake|Monday). The latter conditional probability is indeed 1, but it is not the one being used in the argument.

 

[Dale]

A big part of solving any problem is recognizing the best formulation of a problem. In this case, the best formulation would be Bayes rule stated in odds form:
$$O(H:T|A) = O(H:T) \frac{P(A|H)}{P(A|T)}$$
Where H is the coin landed heads, T is the coin landed tails, A is Beauty is awakened during the experiment (i.e. with amnesia, being interviewed, and being asked her credence that it is heads).

Since the coin is fair O(H:T) is 1 (1:1 odds). The conditional probability P(A|H) is strange, but it is actually not important. What is important is the ratio of P(A|H)/P(A|T), which is clearly 1/2. So then O(H:T|A) = 1/2 (2:1 odds against H), which is a conditional probability of 1/3 for H.

The information is clear, the prior is clear, and the day of the week doesn't even need to enter into the calculation.

 

[PeterDonis]

What is important is the ratio of P(A|H)/P(A|T), which is clearly 1/2.

Ah, I see. The day of the week only plays an indirect role, in deriving this ratio, and the conditional probability for it being a particular day of the week does not enter into it. I agree this is a much cleaner formulation for the odds. However, I'm wondering how you would compute the expected payoff for bets in this way in a scenario like the second one in post #67 (where only one bet is paid off even if Beauty is awakened twice).

 

[Boing3000]

This is not correct. P(Monday|Awake) is the conditional probability that it is Monday, given that Beauty is awake. It is not the conditional probability that Beauty is awake, given that it is Monday; that would be P(Awake|Monday). The latter conditional probability is indeed 1, but it is not the one being used in the argument.

I agree, but my statement is : the drug only use, in the context in this exact scenario, is to make P(Monday) = 1 => P(Monday|Anything) = 1 (anything including no coin flipped at all)
If not, there is no reason to put beauty to sleep and ask her anything. She can answer everything Sunday evening, and do something better with her live, because no information appears past Sunday evening. I cannot get Dale or Steven to acknowledge that.

 

[Dale]

But just knowing the probabilities isn't enough. You also have to know the payoffs attached to each possible outcome. In other words, you are betting on the weighted expectation of the payoff.

Yes, this is true. What I mean is that a rational assessment of the probability of an event must be the same probability as used to calculate the payoff odds at which the rational assessor would be indifferent to a wager on that event occurring.

 

[PeterDonis]

the drug only use, in the context in this exact scenario, is to make P(Monday) = 1 => P(Monday|Anything) = 1 (anything including no coin flipped at all)

I don't understand. What do you mean by P(Monday) = 1 => P(Monday|Anything) = 1?

 

[stevendaryl]

You could simple have answered the question "explain your statement that she can test her amnesia or "update her information", instead of running in circle a get back to this formula which does not apply to the Sleeping Beauty problem.

So you are rejecting a standard formula of the theory of probability. That's what I thought, but you earlier denied rejecting probability theory.

 

[PeterDonis]

What I mean is that a rational assessment of the probability must be the same probability as used to calculate the payoff odds at which the rational assessor would be indifferent.

Yes, but which odds are those? That will depend on how the payoffs are structured. In the two scenarios described in post #67, the payoffs are structured such that different odds are relevant:

In the first scenario, where all bets made are paid off at the end, the relevant odds are O(H:T|A).

In the second scenario, where only the last bet made is paid off at the end, and any other bets made are discarded, the relevant odds are just O(H:T).

 

[Dale]

However, I'm wondering how you would compute the expected payoff for bets in this way in a scenario like the second one in post #67 (where only one bet is paid off even if Beauty is awakened twice).

That is actually a malicious wager, and I don't know how to compute odds with malicious agents who refuse to honor certain bets. I would suspect that such a computation should come out to 1/2 in this case, but that is a gut feeling with no analysis behind it.

But that is not the scenario considered in the Wikipedia description where Beauty is asked her credence "now".

 

[Boing3000]

I don't understand. What do you mean by P(Monday) = 1 => P(Monday|Anything) = 1?

I mean: no recollection of monday while having recollection of sunday make you certain that your are monday.
That's the world Beauty is in. The opinions of outsider are useless to her.

 

[stevendaryl]

I mean: no recollection of monday while having recollection of sunday make you certain that your are monday.
That's the world Beauty is in. The opinions of outsider are useless to her.

That's clearly wrong. The fact that she doesn't know whether it is Monday or Tuesday does not imply that it is Monday.

 

[stevendaryl]

And again, P(Monday | awake) = 1, because of the drug.

Do you really believe that? Sleeping Beauty knows the rules of the experiment. So she knows upon wakening that there are three possibilities:

1. It is monday, and the coin flip result was heads
2. It is monday, and the coin flip result was tails
3. It is tuesday, and the coin flip result was tails.

You're saying that her conclusion is: Today must be Monday?

 

[PeterDonis]

no recollection of monday while having recollection of sunday make you certain that your are monday

No, it doesn't, because she is in exactly the same position on Tuesday, since the drug erases her recollection of Monday.

 

[Boing3000]

However, I'm wondering how you would compute the expected payoff for bets in this way in a scenario like the second one in post #67 (where only one bet is paid off even if Beauty is awakened twice).

I have modified the code to simulate post#67 setup. Second parameter is Beauty strategy likelihood to bet on tail

(function BeautyBets67(run, beautyGuess, betAmount, rule) {
    var counts = { Wins: 0 }
    var experiments = [
            function headBet(counts) {
                var headBet = Math.random() >= beautyGuess;
                if (headBet)
                    counts.Wins += betAmount;
                else
                    counts.Wins -= betAmount;
            },
            function Tail(counts) {
                var tailBet = Math.random() < beautyGuess;
                if (tailBet)
                    counts.Wins += betAmount;
                else
                    counts.Wins -= betAmount;

                if (rule !== "B") {
                    if (tailBet) {
                        counts.Wins += betAmount;
                    }
                    else {
                        counts.Wins -= betAmount;
                    }
                }
            }
    ];
    while (--run >= 0) {
        experiments[Math.floor(Math.random() * experiments.length)](counts);
    }
    alert("Money " + counts.Wins);
})(10000, 0.5, 100, "A");

 

[Boing3000]

No, it doesn't, because she is in exactly the same position on Tuesday, since the drug erases her recollection of Monday.

Which is exactly the reason why she is drugged, to make her certain that she is Monday, which is the exact same position (your words)

 

[stevendaryl]

Ok, but we should really start by defining the "probability spaces" that are involved.

That's the challenge: to figure that out.

The original problem mixes up two different notions of possibility (which maybe is the source of the confusion?):

1. There are alternate "possible worlds": One where the coin flip result is heads, and one where the coin flip result is tails.
2. There are alternate days within one world.

(awake versus asleep is a function of the other two variables). So there are 4 possibilities:

1. (tails, monday, awake)
2. (tails, tuesday, awake)
3. (heads, monday, awake)
4. (heads, tuesday, asleep)

From Sleeping Beauty's point of view, she wakes up and knows that she is in situations 1-3 (she can't actually experience #4). She is uncertain about what day it is, and she is uncertain about what the coin flip result was. The point of the problem is to come up with a sensible way to quantify these uncertainties. So from a Bayesian reasoning point of view, she's being asked to come up with sensible prior probabilities.

 

[stevendaryl]

Which is exactly the reason why she is drugged, to make her certain that she is Monday, which is the exact same position (your words)

The point of the drug is so that she can't tell the difference between Monday and Tuesday. It's not to fool her into thinking it's Monday when it isn't. She knows that it's possible that she's been drugged.

There is a difference between not knowing what day it is and falsely believing that it is Monday when it's not.

 

[Dale]

Please tell me how YOU would KNOW that you have amnesia, after waking up

I would know because I would recall volunteering for the experiment, I would recall that induced amnesia is part of the experimental protocol, I would not recall awakening after Sunday, and the interviewer would be interviewing me according to the protocol.

 

[Boing3000]

That's clearly wrong. The fact that she doesn't know whether it is Monday or Tuesday does not imply that it is Monday.

No ? In my book, every time I have only recollection of the previous day, then I am the next day. Unless I am in a hospital, or with evil mathematician torturing my for NO REASON.

Do you really believe that? Sleeping Beauty knows the rules of the experiment. So she knows upon wakening that there are three possibilities:

Of course she knows it. She knows that she might be drugged. That's not what she is asked about.
The only thing she know for a fact, is that she is Monday. This is the only information she have that is different from the lab guys.

You're saying that her conclusion is: Today must be Monday?

I am saying here conclusion is not based on coin value, that she has no access to.
Nor ii is based on yesterday calculation that will never be updated.

She is asked about her credence about that coin now. And her now is forever Monday. There is no other reasons to drug her. Not one that you can come up with anyway, to justify your computation.

 

[Dale]

Actually, what you know in the Sleeping Beauty problem is that you have no memory of Monday. That could be because of amnesia, or it could be because today is Monday.

That is a good point, and is a better way of stating the situation. What she does know is that she is being awakened as part of the study protocol. She knows that, as you say, and from her agreement to enter the study.

 

[Boing3000]

I would know because I would recall volunteering for the experiment, I would recall that induced amnesia is part of the experimental protocol, I would not recall awakening after Sunday, and the interviewer would be interviewing me according to the protocol.

In other words you only have probability based on prior knowledge. I asked you about a way to test your amnesia.

 

[Dale]

In other words you only have probability based on prior knowledge. I asked you about a way to test your amnesia.

That is all that is needed for probabilistic inference. Indeed, that is all you get from any medical test or scientific experiment

 

[PeterDonis]

In my book, every time I have only recollection of the previous day, then I am the next day.

In other words, you are defining "I can't remember anything past Sunday" as being equivalent to "it is Monday". Which is not the way anyone else in this discussion is defining "it is Monday". Everyone else is defining "it is Monday" as it actually being Monday, regardless of anyone's memory or lack thereof; thus, according to the way everyone else is using language, "I can't remember anything past Sunday" is consistent with either "it is Monday" or "it is not Monday, it is Tuesday, but my memory of Monday has been erased by the drug".

I suppose this is an extreme example of the vagueness of ordinary language. But in your idiosyncratic use of language, we would still need to somehow distinguish the cases "I can't remember anything past Sunday because it is actually Monday" and "I can't remember anything past Sunday because it is actually Tuesday and my memory of the actual Monday has been erased". So how, in your use of language, would you distinguish those cases? Once you answer, then just go back and substitute your answer everywhere that anyone except you says "it is Monday" or "it is Tuesday", and so forth. Then you will be talking about the same actual math as the rest of us.

 

[stevendaryl]

No ? In my book, every time I have only recollection of the previous day, then I am the next day. Unless I am in a hospital, or with evil mathematician torturing my for NO REASON.

She's told the rules ahead of time. She knows that there is a possibility of it being Monday or Tuesday, because that's the way the experiment was set up.

She is asked about her credence about that coin now. And her now is forever Monday.

That makes no sense. The fact that she doesn't know whether it's Monday or Tuesday doesn't mean that it's Monday.

 

[Boing3000]

That is all that is needed for probabilistic inference. Indeed, that is all you get from any medical test or scientific experiment

You don't need to be put to sleep for that. You don't need to be drugged for that. Beauty problem is not about generic and easy probabilistic inference.
That problem contains a specific procedure to break that inference.

Or maybe you could explain why she has to be put to sleep or drugged randomly ?
What if we use a drug to implant memories of Monday. Does it help ?

 

[Dale]

Which is is case for EVERY day. This does NOT change

It certainly does change. Specifically it is not the case on Sunday or earlier and it is not the case on Wednesday or later. Her observations let her know that she is being awoken as part of the experimental protocol on Monday or Tuesday. This is information.

 

[Dale]

Beauty problem is not about generic and easy probabilistic inference.
That problem contains a specific procedure to break that inference.

The procedure is designed so that Beauty cannot condition on Monday or Tuesday, but she can still marginalize over them. (Although even that is not necessary)

 

[PeterDonis]

The only thing she know for a fact, is that she is Monday.

But only on your idiosyncratic definition of "it is Monday". That is, she knows for a fact that she can't remember past Sunday. But she does not know for a fact that it is actually Monday, the way everyone but you uses that term. For example, she does not know for a fact that if she were to ask the experimenter who has just awakened her "what day is it?", he would answer "Monday". Or that if she looked at a computer-driven clock/calendar that always showed the current day of the week according to local time, it would say the day of the week was Monday. Or, etc., etc.

 

[Boing3000]

Everyone else is defining "it is Monday" as it actually being Monday,

We are not discussing time. There is no calendar in the room. I am sticking to the exact and precise explicit data that is found in the definition of the problem.

regardless of anyone's memory or lack thereof;

That is quite wrong. Time is a continuum, and memories is the only human clock.

thus, according to the way everyone else is using language, "I can't remember anything past Sunday" is consistent with either "it is Monday" or "it is not Monday, it is Tuesday, but my memory of Monday has been erased by the drug".

Or we are a Friday in year 2859 and I have taken a wormhole.
The way people connect event is the same as clock connect second. Monday is what follow Sunday.

I suppose this is an extreme example of the vagueness of ordinary language. But in your idiosyncratic use of language, we would still need to somehow distinguish the cases "I can't remember anything past Sunday because it is actually Monday"

A would like to see a non-idiosyncratic calendar where Monday popup randomly instead of every single time after Sunday.

and "I can't remember anything past Sunday because it is actually Tuesday and my memory of the actual Monday has been erased". So how, in your use of language, would you distinguish those cases?

The one (that have been drugged) cannot distinguish those two, which is the purpose of this explicit criteria.

Once you answer, then just go back and substitute your answer everywhere that anyone except you says "it is Monday" or "it is Tuesday", and so forth. Then you will be talking about the same actual math as the rest of us.

I have no problem with the math of the rest of you. You have problem understanding the problem of the problem.
There is no math covering time travel backward in time with probability attached.

 

[stevendaryl]

We are not discussing time. There is no calendar in the room. I am sticking to the exact and precise explicit data that is found in the definition of the problem.

Then you are arguing about a different thought experiment than the rest of us are.

 

[Boing3000]

The procedure is designed so that Beauty cannot condition on Monday or Tuesday, but she can still marginalize over them. (Although even that is not necessary)

So the condition left is she is in a room (awake, because some here suppose is is even relevant), you are making my points.
If the procedure does not change anything to her actual way to give credence, what's the point again?

 

[Boing3000]

Then you are arguing about a different thought experiment than the rest of us are.

I 'll let you bicker another half century with you pairs, about such a simple problem which allow so many solution.
OK, I have been told enough those kind of nonsense. My idiosyncratic use of language does not allow me to understand the following.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before

 

[stevendaryl]

So the condition left is she is in a room (awake, because some here suppose is is even relevant), you are making my points.
If the procedure does not change anything to her actual way to give credence, what's the point again?

Sleeping Beauty wakes up, and the experimenter tells her that she is in one of three situations:

1. His coin flip (hidden from her) resulted in heads, and today is Monday.
2. His coin flip resulted in tails, and today is Monday.
3. His coin flip resulted in tails, and today is Tuesday, and her memory of Monday has been erased.

She knows that those are the possibilities. She doesn't know which one is her actual situation.

Everybody else has known from the start of this thread that those were the possibilities, and that Sleeping Beauty was aware of those possibilities. If you want to talk about a different situation, where she believes that today is definitely Monday (even if it's not), fine. But that should be a different thread.

 

[stevendaryl]

OK, I have been told enough those kind of nonsense. My idiosyncratic use of language does not allow me to understand the following.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before

That means that she doesn't know whether it is Monday or Tuesday. It doesn't mean that she believes that it is Monday.

 

[Boing3000]

Everybody else has known from the start of this thread that those were the possibilities, and that Sleeping Beauty was aware of those possibilities.

Actually you were not aware of them before I had to explains them to you.

If you want to talk about a different situation, where she believes that today is definitely Monday (even if it's not), fine. But that should be a different thread.

I talk about the explicit situation, while you are not.

 

[Dale]

So the condition left is she is in a room (awake, because some here suppose is is even relevant),

The condition is that she was awoken in a room as part of the study protocol. That condition is correlated with Heads.

 

[PeterDonis]

The one (that have been drugged) cannot distinguish those two, which is the purpose of this explicit criteria.

Once again, you are using language in a highly idiosyncratic way. To you, "cannot distinguish those two" means "the first of the two must be true". To the rest of us, "cannot distinguish those two" means "must allow for the possibility of both". And the latter is what is specified in the problem. So, once again, you need to rewrite the problem statement in your idiosyncratic language so that you are talking about the same thing as the rest of us.

 

[Boing3000]

That means that she doesn't know whether it is Monday or Tuesday. It doesn't mean that she believes that it is Monday.

What you believe she believe is irrelevant to the situation. She is not asked about here believe to be Monday, nor is it relevant, because she is, as far as she can tell.
What new information she discover after awakening which is different from Sunday evening is relevant.
50 posts down the drain, and I am still waiting an answer...

 

[PeterDonis]

And the latter is what is specified in the problem.

Unless, I guess, we want to look at this as an even more extreme example of the vagueness of ordinary language.

 

[PeterDonis]

50 posts down the drain, and I am still waiting an answer...

Read my post #290 (and read #281 again). The answer is that you are insisting on interpreting language in a way that is very, very different from everyone else. I can't say that your interpretation is "wrong", since I have made the point multiple times in this thread that ordinary language is vague; but you also can't insist that your interpretation is "right" and everyone else's is "wrong". So the answer that you say you are waiting for has already been given to you, multiple times now: you are answering a different question than the rest of us, and you don't seem to even comprehend the possibility that the interpretations the rest of us are putting on the ordinary language in the problem statement are valid interpretations.

 

[stevendaryl]

What you believe she believe is irrelevant to the situation. She is not asked about here believe to be Monday, nor is it relevant, because she is, as far as she can tell.

You don't seem to understand how conditional probability works. If you don't know whether today is Monday or Tuesday, and someone asks you what the probability of X is, then you use conditional probability formula:

P(X) = P(X | Monday) P(Monday) + P(X | Tuesday) P(Tuesday)

So you don't need to know whether it is Monday or Tuesday, but you need to know what the probabilities of it being Monday versus Tuesday are.

Your approach seems to be: I don't know whether it is Monday or Tuesday. So I'll just assume it is Monday.

 

[stevendaryl]

What new information she discover after awakening which is different from Sunday evening is relevant.
50 posts down the drain, and I am still waiting an answer...

The fact that she's awake is itself information! Consider a different problem where if the coin is heads, she isn't wakened at all. Then do you agree that upon being awakened, she will know that the coin flip was tails?

 

[Boing3000]

To you, "cannot distinguish those two" means "the first of the two must be true".

I certainly don't do that. She certainly can distinguish Monday from Tuesday with the explicit apparatus called a brain, with her explicit memory erase (which is explicitly identical to no having those memory yet)

To the rest of us, "cannot distinguish those two" means "must allow for the possibility of both".

That's lab frame information, no relevant to putting her to sleep.
Why not trying to explain what this "putting into a sleep" means in mathematics language, and how it will change here credence computable the Sunday evening ..
That would certainly be more useful than insulting people.

And the latter is what is specified in the problem. So, once again, you need to rewrite the problem statement in your idiosyncratic language so that you are talking about the same thing as the rest of us.

Nobody can because the problem statement is irrational, and that "the rest of us" is split in 3 "camps" at least.
You should be aware that you being wrong does not make everyone else right.

 

[Dale]

What new information she discover after awakening which is different from Sunday evening is relevant

She discovers that she is currently in the trial, which is also when she is asked to assess her credence.

 

[stevendaryl]

I certainly don't do that.

You said it many times: For Sleeping Beauty, it's always Monday.

Maybe you didn't mean it, but what do people have to go on, other than what you say?

Nobody can because the problem statement is irrational, and that "the rest of us" is split in 3 "camps" at least.
You should be aware that you being wrong does not make everyone else right.

The problem statement is perfectly clear: Sleeping Beauty wakes up knowing that she is in one of three situations:
(1) Monday and Heads, (2) Monday and Tails, (3) Tuesday and Tails. She knows that if she were in situation #3, that means that her memory of what happened on Monday was erased. She doesn't know which of the three is the case, but she's being asked to quantify her uncertainty by giving a subjective likelihood that she's in situation #1.

There might be multiple plausible answers, but the problem statement is clear enough.

 

[Boing3000]

You don't seem to understand how conditional probability works. If you don't know whether today is Monday or Tuesday, and someone asks you what the probability of X is, then you use conditional probability formula:

P(X) = P(X | Monday) P(Monday) + P(X | Tuesday) P(Tuesday)

That the 10 times now that I explicitly say that this the lab knowledge compute Sunday evening. Do you copy ?

Your approach seems to be: I don't know whether it is Monday or Tuesday. So I'll just assume it is Monday.

That's not my approach. That's the one explicitly described in the article as the only reason to change her credence between Sunday evening and awakening.
I am still waiting your other alternative explanation.

The fact that she's awake is itself information

No it's not. She cannot update any likelihood/credence with this information. I am just sorry you cannot understand that.

 

[stevendaryl]

No it's not. She cannot update any likelihood/credence with this information. I am just sorry you cannot understand that.

Because it's false! If heads versus tails changes the number of times she is awakened, then the fact that she is awake changes the subjective likelihood that it's heads or tails. That's the way conditional probability works.