Undergrad The Sleeping Beauty Problem: Any halfers here?

[Boing3000]

When she wakes up in the morning, the fact that she doesn't know whether it's Monday or Tuesday tells her something. It tells her that either it's Monday, or that the coin toss was (whichever one results in a memory wipe).

It tells here NOTHING more that she knew already. You cannot understand the problem of that problem until you acknowledge that.

But that's not true. She doesn't cease to exist on Tuesday in either case. It's just that if it's Tuesday, and the coin toss was (whichever result does not result in a memory wipe), then she will know what day it is.

1) Fact : the coins can be tossed Monday night. Apparently you don't get that.
2) Fact : She factually cease to exist; From wiki "Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening".
Thus I beg to disagree with the underlined part, because it is simply wrong.

Therefore, if she doesn't know what day it is, then she learns either that it's Monday, or that it's Tuesday and her memory has been wiped.

So she learned nothing. She knew that already. And she have definitely no way two distinguish between those "OR".

It certainly doesn't do the halfers any good to justify their answer by rejecting the whole concept of probabilistic reasoning.

I have not seen any halfers do that. Nor does I. It certainly does not do you any good to pretend that.

Thanks for the "probabilistic inference" reminder. But you need to understand halfers use the same rule, with a different inference (void for Beauty).

 

[stevendaryl]

So she learned nothing. She knew that already.

No, she didn't. If she has memories of Monday, then she learns something (namely, that her memory wasn't wiped, and the experiment is over). If she doesn't have memories of Monday, then she also learns something (namely, that the experiment is not yet over).

 

[stevendaryl]

I have not seen any halfers do that. Nor does I. It certainly does not do you any good to pretend that.

You said yourself that the probability of heads can never be updated in light of new information. Or at least, you seemed to be denying that. You said:

Probabilities of coins flipping does not update with knowledge.

That's contrary to standard probabilistic reasoning.

 

[Boing3000]

No, she didn't. If she has memories of Monday, then she learns something (namely, that her memory wasn't wiped, and the experiment is over). If she doesn't have memories of Monday, then she also learns something (namely, that the experiment is not yet over).

She never have memories of Monday, per definition of that problem. She never forget Sunday, per definition of that problem. Hence she is forever on Monday.
If you cannot understand that from the problem definition, I cannot help you.

You said yourself that the probability of heads can never be updated in light of new information. Or at least, you seemed to be denying that

Nope I never said that. I explicitly said new information AND new correlation is needed.
You keep projecting your way of thinking into mine. I don't use implicit inferences. I am an explicit kind of guy.

Probabilities of coins flipping does not update with knowledge.

See ? There is no new correlation here. Observing coins being flipped will never change that probability.
Knowledge meant data-collection, not purposely asserting new correlation connection. Those are two different things.
And Beauty has no way to access to new information past Sunday.

 

[stevendaryl]

She never have memories of Monday, per definition of that problem.

No, we don't kill her based on the coin flip. It's just that if the coin flip is a certain result, then her memory of Monday is erased. If the coin flip is something else, her memory of Monday is not erased. So she learns something from the fact that her memory is or is not erased.

 

[stevendaryl]

Nope I never said that. I explicitly said

Well, what you said and what you denied saying sound the same to me. I can't make any sense of your claim:

Probabilities of coins flipping does not update with knowledge.

The only kind of updating that is being discussed is revising the probability that a coin flip was heads based on new information. Your quote seems to be saying that that's impossible, which is contrary to probabilistic reasoning.

 

[stevendaryl]

I'm beginning to think that the Sleeping Beauty problem as stated is misleading halfers. The statement of the problem suggests that there are only three possibilities:

1. Heads and it's Monday
2. Tails and it's Monday
3. Tails and it's Tuesday

That's not true. We don't kill Sleeping Beauty depending on the coin toss. She'll wake up Tuesday morning, regardless of the coin flip. The difference is that if the coin flip was "heads", then on Tuesday, she'll know that it is Tuesday, and that the experiment is over. So really, she wakes up every morning. It's just that only under certain circumstances is she uncertain about what day it is. So the fact that she is uncertain about the day tells her something. She does learn something upon awakening.

 

[PeterDonis]

I'm beginning to think that the Sleeping Beauty problem as stated is misleading halfers.

The problem as stated is not what you are stating.

We don't kill Sleeping Beauty depending on the coin toss. She'll wake up Tuesday morning, regardless of the coin flip.

That's not the way the problem is stated. (It's the way you have altered it in your posts, but it's not the way the original problem is stated.) As the original problem is stated, Beauty is only awakened on Tuesday if the coin flip turns up tails. Otherwise she is left asleep on Tuesday. That's how it is described in the Wikipedia article linked to in the OP.

So really, she wakes up every morning.

Only in your altered version of the experiment. Which, once again, illustrates the point I made when I first entered this thread, that the fact that everyone seems to need to alter the definition of the experiment in order to derive a clear answer indicates that the original experiment is not defined precisely enough for there to be a clear answer.

 

[stevendaryl]

The problem as stated is not what you are stating.
That's not the way the problem is stated. (It's the way you have altered it in your posts, but it's not the way the original problem is stated.) As the original problem is stated, Beauty is only awakened on Tuesday if the coin flip turns up tails. Otherwise she is left asleep on Tuesday. That's how it is described in the Wikipedia article linked to in the OP.

So your interpretation is that she sleeps for the rest of eternity in that case? I just interpreted that in that case, the experiment ends and she wakes normally (with an alarm clock, or whatever).

 

[PeterDonis]

So your interpretation is that she sleeps for the rest of eternity in that case?

No, she is awakened on Wednesday and the experiment ends. Read the Wikipedia article linked to in the OP.

 

[Charles Link]

Which, once again, illustrates the point I made when I first entered this thread, that the fact that everyone seems to need to alter the definition of the experiment in order to derive a clear answer indicates that the original experiment is not defined precisely enough for there to be a clear answer.

In many physics problems, (such as problems involving masses and springs and pulleys, etc.), it is useful to test the limiting cases to help solidify that you have the correct answer. In this particular problem, most of the checks on the answer using limiting cases (such as using a biased coin) gave results that brought into question the correctness of the answer that was computed. Although I voted for 1/2, (because the 1/3 result didn't pass the test of the biased coin), I'm not particularly pleased with the 1/2 answer either. Perhaps the 3rd choice is the best of the three, but the problem statement itself seems to be reasonably clear. On first reading of it, I would have expected it to give an answer that passed all the tests.

 

[stevendaryl]

No, she is awakened on Wednesday and the experiment ends. Read the Wikipedia article linked to in the OP.

Yes, I screwed up in stating the problem. I apologize. I was using a different variation of the problem that was not the one from Wikipedia.

I don't think that the answer changes, though. In the Wikipedia article, if the coin toss is heads, then Sleeping Beauty is only awakened once. If we changed that to "if the coin toss is heads, she is not awakened at all", then surely the odds of the coin toss being tails given that she is awake must rise to 1. So the argument that she learns nothing from the fact that she is awake isn't true, as a general principle.

 

[PeroK]

In many physics problems, (such as problems involving masses and springs and pulleys, etc.), it is useful to test the limiting cases to help solidify that you have the correct answer. In this particular problem, most of the checks on the answer using limiting cases (such as using a biased coin) gave results that brought into question the correctness of the answer that was computed. Although I voted for 1/2, (because the 1/3 result didn't pass the test of the biased coin), I'm not particularly pleased with the 1/2 answer either. Perhaps the 3rd choice is the best of the three, but the problem statement itself seems to be reasonably clear. On first reading of it, I would have expected it to give an answer that passed all the tests.

I have to say that your result with the biased coin must have been due to a calculation error.

 

[Boing3000]

The only kind of updating that is being discussed is revising the probability that a coin flip was heads based on new information

And I am telling you that new information does NOT change anything (learning that "my tailor is rich" is irrelevant, and it IS new information).
You need new correlation, and then new data concerning this new sets of possible facts. You need to be specific.

If we changed that to "if the coin toss is heads, she is not awakened at all"

You don't get to change the correlation between events mid-way analyzing a problem.

then surely the odds of the coin toss being tails given that she is awake must rise to 1. So the argument that she learns nothing from the fact that she is awake isn't true, as a general principle.

You cannot use fallacies to derive general principle. That she cannot learn anything is a basis for that "dilemma".
It is easy to spot, you just cannot come to term that this problem is badly defined (but convincingly enough apparently) so that physicists/mathematicians start trolling themselves to oblivion.

 

[Charles Link]

I have to say that your result with the biased coin must have been due to a calculation error.

With the biased coin, where the coin is 99% biased for heads, but also where the tails result is changed to waking her for 1000 consecutive days gives a mathematical result that defies logic. There is only one coin flip, and not an ensemble...

 

[PeroK]

With the biased coin, where the coin is 99% biased for heads, but also where the tails result is changed to waking her for 1000 consecutive days gives a mathematical result that defies logic. There is only one coin flip, and not an ensemble...

Sorry, that just makes no sense.

 

[PeterDonis]

I don't think that the answer changes, though.

Basically, this type of claim (of which a number of different individual examples have been made in this thread) amounts to saying that the answer to the original problem is the same as the answer to some different problem that is more precisely specified. Which amounts to saying that the original problem can only be interpreted such that it is equivalent to the more precisely specified problem. But this thread discussion clearly falsifies that claim, since different people are interpreting the original problem specification differently.

 

[PeroK]

Only in your altered version of the experiment. Which, once again, illustrates the point I made when I first entered this thread, that the fact that everyone seems to need to alter the definition of the experiment in order to derive a clear answer indicates that the original experiment is not defined precisely enough for there to be a clear answer.

In my summary post, a few pages back now, I stuck to the problem,as stated. In fact, early on in this thread the claim was made that all thirder arguments involve changing the problem in some way. Since then I have stuck to the core problem as stated.

My only embellishment is to consider an additional random observer, but that it simply to corroborate Beauty's calculations, which could be left to stand without corroboration.

I have done this despite the fact that I believe increasing the number of days involved blows the 1/2 argument apart. The halfers rely on the fact that both 1/2 and 1/3 are plausible answers.

With, say, 99 days, the options would be 1/2 or 1/100. The 1/2 answer becomes very difficult to justify in this case. It becomes increasingly implausible.

In fact, the other difficulty with the 1/2 answer is that it becomes the general answer for a wide range of data; whereas, the 1/3 answer changes with the numerical data involved, which is more what you would expect.

The exception to this is your argument that the problem might imply simply a repeat of the calculation with the original data. In that case it makes perfect sense that 1/2 is the only answer no matter what has happened or may yet happen.

That, however, renders the problem rather pointless, implying as is does simply a repeat of the same calculation in all circumstances.

It's interesting that some of the wilder defences of the 1/2 position seem to have taken this to the extreme that even if you look at the coin and see a head, say, then the question over the probability that it is heads remains 1/2. And that the certainty now that it is a head is not actually a probability.

This also leads to the extreme position that the probability of a coin being heads is absolutely 1/2 and can never change, even to 0 or 1, under any circumstances.

These reveal fundamental disagreements over the nature of probability and credence that wouldn't be removed even if the problem were stated more precisely, as you suggest.

 

[PeterDonis]

In my summary post, a few pages back now, I stuck to the problem,as stated...

My only embellishment is...

In other words, you stuck to the problem as stated, except for embellishments. Which just concedes my point.

that it simply to corroborate Beauty's calculations, which could be left to stand without corroboration

Only if those calculations are the ones that correspond to the vague ordinary language in the original problem statement. Which you argue for by adding embellishments. Which again concedes my point.

These reveal fundamental disagreements over the nature of probability and credence that wouldn't be removed even if the problem were stated more precisely, as you suggest.

Yes, that's true. In other words, the original problem statement is vague, and making it precise requires picking one set of assumptions about the nature of probability and credence, out of multiple mutually incompatible possibilities of which different people will pick different ones.

 

[stevendaryl]

Basically, this type of claim (of which a number of different individual examples have been made in this thread) amounts to saying that the answer to the original problem is the same as the answer to some different problem that is more precisely specified.

The whole challenge of thought experiments such as this one is to figure out how to cast the problem into a form where mathematics can be applied. There are two issues in reformulating it: (1) Resolving ambiguities in the problem that don't show up until you try to formalize it. (2) Figuring out what is the correct way to formalize it, in the first place.

I don't think those are the same thing. The fact that people can formalize it two (or more) different ways does not mean that the problem was ambiguous. It might mean that some of the ways of translating an informal problem into a mathematical one are just wrong.

Which amounts to saying that the original problem can only be interpreted such that it is equivalent to the more precisely specified problem.

That's what I think.

But this thread discussion clearly falsifies that claim, since different people are interpreting the original problem specification differently.

It could be a matter of interpreting it differently, or it could be a matter of some people making a conceptual mistake. If you have a problem, such as the Bell Spaceship paradox, and people come up with different answers, that doesn't mean that the problem was ambigous. It sometimes means that somebody made a mistake in formalizing. You can't use the fact that people get different answers as an argument that the problem was ambiguous. (By that criterion, every answer is always right, because the fact that someone got a different answer is proof that they interpreted the problem differently.)

 

[PeterDonis]

It might mean that some of the ways of translating an informal problem into a mathematical one are just wrong.

It might. Or it might mean the informal problem was ambiguous. In the end that's a judgment call.

 

[stevendaryl]

Yes, that's true. In other words, the original problem statement is vague, and making it precise requires picking one set of assumptions about the nature of probability and credence, out of multiple mutually incompatible possibilities of which different people will pick different ones.

That's in the nature of thought experiments. That's sort of the point of them---you have an informally described situation, and the challenge is to formalize them sufficiently well that are amenable to being solved through mathematics, logic and possibly experiment. The fact that different people choose different formalizations does not by itself mean that all ways of formalizing it are equal. Some ways of formalizing it just don't stand up to scrutiny.

That's sort of the point of exploring slight tweaks of the problem statement. If a principle of reasoning is used in the original problem, and applying the same principle in the tweaked case clearly gives the wrong answer, that suggests that the principle of reasoning is flawed.

It's certainly possible that such a problem is inherently ambiguous; there are multiple, completely legitimate interpretations that lead to different answers. But just the fact that someone proposes a solution doesn't automatically make it correct.

 

[stevendaryl]

It might. Or it might mean the informal problem was ambiguous. In the end that's a judgment call.

Right. To me, the test that a solution is a legitimate interpretation is precisely if the solution approach is robust with respect to slight tweaks of the problem.

 

[Boing3000]

It's interesting that some of the wilder defences of the 1/2 position seem to have taken this to the extreme that even if you look at the coin and see a head, say, then the question over the probability that it is heads lands heads when flipped remains 1/2.

I hope that your are not referring to my posts, first because I don't defend 1/2 over 1/3 because they are both valid.
Secondly because, as stevendaryl, you insert error of omission, that I corrected above in bold
In this case the probability is associated with the action not with a state, or lack of information about that state.

And that the certainty now that it is a head is not actually a probability.

I'll let you slip hairs further to decide if certainty equals 1.0 probability.
What I am sure is that a coins once landed on head, never ever turn into a tail, nor became a bowl of petunia. In my book it is not a probability.

This also leads to the extreme position that the probability of a fair coin flipped, will land on being heads is absolutely 1/2 and can never change, even to 0 or 1, under any circumstances.

Coin never are in any other states than 0 or 1. There is a 1/2 possibility that you observe it being tail or head.
There are even circumstances when flipping is unfair, and nobody said otherwise. You can very much train yourself or a robot with precise mechanics to approach 0 or 1 probability when flipped expertly (not flipped "fairly"), even if the coin itself is fair. That's what knife throwers do (with sharp coins).
You can also build any type of dependencies between a series of events leading in all types of probabilities. This is basic knowledge.

These reveal fundamental disagreements over the nature of probability and credence that wouldn't be removed even if the problem were stated more precisely, as you suggest.

I have seen no such fundamental disagreements.

 

[stevendaryl]

I think that there is a related problem to the Sleeping Beauty problem whose solution is implicit in any solution of the Sleeping Beauty.

Suppose that Sleeping Beauty is awakened with probability \alpha on Monday, and with probability \beta on Tuesday (after a memory wipe). What is her subjective probability that today is Monday, given that she's awake?

I claim that it is \frac{\alpha}{\alpha + \beta}. You can justify that using relative frequency: If you do it over and over again, N times (once a week for N weeks) then she'll be awake approximately (\alpha + \beta) N times, and of those, \alpha N will be Mondays. So the relative frequency of her being awake on Mondays as a fraction of the total times that she is awake is \frac{\alpha}{\alpha + \beta}.

If halfers have some other answer, then please say what it is. I'm assuming that the only other possible answer is: Undefined. However, that doesn't seem right, because in the case \alpha = 0, then Sleeping Beauty knows for sure that it's Tuesday, and if \beta = 0, then she knows for sure that it's Monday.

 

[PeterDonis]

What is her subjective probability that today is Monday, given that she's awake?

Yes, this is what I was calling P(Monday|Awakened) in earlier posts. Your formula gives the answer 2/3. I argued in a previous post that this answer can only be justified (based on relative frequencies) if Beauty is told that the experiment will be run a large number of times, and she will be given the amnesia drug at the end of each run so that during each run she doesn't know how many runs have been made. For a single run, I argued that a maximum entropy prior could be used, which gives the answer 1/2 (and therefore the answer 1/4 to the question of what P(Heads) is, which would I guess be a "quarterer" answer).

This line of argument does seem to make the halfer position hard to maintain, since by the equations in my earlier post, P(Heads) = 1/2 requires P(Monday|Awakened) = 1, i.e., Beauty would need to be certain that it was Monday when she was awakened.

My relevant earlier posts are here:

https://www.physicsforums.com/threa...m-any-halfers-here.916459/page-7#post-5780588

https://www.physicsforums.com/threa...m-any-halfers-here.916459/page-8#post-5781206

 

[PeroK]

In other words, you stuck to the problem as stated, except for embellishments. Which just converse my point.

Only if those calculations are the ones that correspond to the vague ordinary language in the original problem statement. Which you argue for by adding embellishments. Which again concedes my point.
.

In this thread, I believe I am representing mainstream mathematical thought. By undermining those of us who are doing this, and saying nothing to counter the tide of personal theorising, you are, I believe, acting against the spirit of PF.

These last comments - effectively denying us any attempt to illuminate a problem in any way and attacking us for doing so are particularly harmful.

I wish I could understand your motivation for attacking only those of who are presenting material in a measured analytical fashion in this thread.

You have lost a great deal of my respect for you.

Sadly, those are my last words on this thread.

 

[PeterDonis]

In this thread, I believe I am representing mainstream mathematical thought.

If "mainstream mathematical thought" includes the frequentist interpretation of probability, then I suppose you are. But, as I said before, not everyone is a frequentist--not even all "mainstream" mathematicians, AFAIK.

the tide of personal theorising

If you think particular posts in this thread are personal theorizing, by all means report them.

attacking

I don't think pointing out assumptions or limitations in a particular line of thought is "attacking". I also don't see what's so difficult about admitting that the original specification of some problem in a Wikipedia article is not very precise (note that I only came into this thread after that option was added to the poll). I'm not saying you personally wrote an imprecise or vague specification of the problem.

You have lost a great deal of my respect for you.

I'm sorry you feel that way. My intention has not been to "attack" anyone, nor to prevent anyone from illuminating the problem.

 

[stevendaryl]

Yes, this is what I was calling P(Monday|Awakened) in earlier posts. Your formula gives the answer 2/3. I argued in a previous post that this answer can only be justified (based on relative frequencies) if Beauty is told that the experiment will be run a large number of times, and she will be given the amnesia drug at the end of each run so that during each run she doesn't know how many runs have been made. For a single run, I argued that a maximum entropy prior could be used, which gives the answer 1/2 (and therefore the answer 1/4 to the question of what P(Heads) is, which would I guess be a "quarterer" answer).

Well, if on Monday you wake Sleeping Beauty with probability \alpha, and on Tuesday you wake her with probability \beta, it seems that her credence for today being Monday would have to depend on \alpha and \beta, at least in the extremes:

\alpha = 0, \beta > 0 \Longrightarrow P(Monday | Awakened) = 0
\alpha > 0, \beta =0 \Longrightarrow P(Monday | Awakened) = 1
\alpha = \beta > 0 \Longrightarrow P(Monday | Awakened) = 1/2

My formula interpolates smoothly between those extremes, even if you don't buy the relative frequency argument in the case of one trial.

 

[Charles Link]

Well, if on Monday you wake Sleeping Beauty with probability \alpha, and on Tuesday you wake her with probability \beta, it seems that her credence for today being Monday would have to depend on \alpha and \beta, at least in the extremes:

\alpha = 0, \beta > 0 \Longrightarrow P(Monday | Awakened) = 0
\alpha > 0, \beta =0 \Longrightarrow P(Monday | Awakened) = 1
\alpha = \beta > 0 \Longrightarrow P(Monday | Awakened) = 1/2

My formula interpolates smoothly between those extremes, even if you don't buy the relative frequency argument in the case of one trial.

Did I misread the problem? I thought that Sleeping Beauty gets awoken Monday, regardless of heads or tails, but for Tuesday only if it's tails.