The Sleeping Beauty Problem: What is the Scientific Definition of Credence?

[Office_Shredder]

This is not the same problem. In this case she gains information when she wakes up since there was a possibility that she wouldn’t wake up at all. So here it’s clear her credence that it landed heads is 1/3.

What about the last part of my post, where we run it multiple times? You get the exact same outcome that we wake her up twice as many times when the coin is tails as it is heads, but do you think her answer should change depending on which of the experiments we are running?

 

[Moes]

Because her belief is expressed by her willingness to accept a bet, and the rational bet is the “thirder” bet. It would be irrational to bet in contradiction to one’s beliefs and it would be irrational to choose a worse bet.

No, the chances are not 50/50. Why would they be?

Since you have proposed an alternate, let me do so as well, we will call this one the “Extreme Edition”:

Suppose that she is told in the beginning that if it is heads she will not be woken and interviewed on either Monday or Tuesday while if it is tails she will be woken and interviewed on Monday and Tuesday. What should be her credence for heads then? Why?

I think you should think through my case again because if you still argue there then I’m miss understanding your argument

The main difference I was making with the second bet was that it is only offered in one awakening whether the coin landed heads or tails.

I thought you agreed the probability of the coin toss remains 50/50. You were just arguing that since betting on tails comes out to her advantage she should for some reason believe in tails more.

By the second bet she is asked whether she thinks she won the first bet or not. The chances of her winning the first bet was 50/50. 50% chance she loses and 50% she wins double.

She is not asked whether she thinks she chose the better option but rather if she thinks she will win the first bet.

I can make a random bet where there is a 50% chance I lose 10 dollars and 50% chance I win 20. The other person may be stupid for agreeing to it but it won’t mean I have a better chance of winning the bet. The money won’t change the 50/50 chance.(I agree the circumstances are different here but for what I’m arguing that doesn’t matter)

So although I agree in the first bet she should choose tails in the second bet she gets no advantage if she chooses tails since she has no chance of winning double. She is only offered the bet once.

Where exactly are you arguing?

 

[Moes]

In the Extreme addition you propose she gains new information by waking up. So her credence for heads is now 0.

 

[Moes]

Do you accept that if this experiment is repeated many times, then the sleeper is woken up twice as often when the coin was tails as when it was heads? Suppose we ran the experiment 100 times and got 50 heads and 50 tails. The experiment in total would last 150 days (2 days every time a tail came up and 1 day every time a head comes up).

The experiment looks like:

Day 1: toss coin; Head; wake sleeper; end of experiment 1
Day 2: toss coin; Tails; wake sleeper
Day 3: wake sleeper; end of experiment 2
Day 4: toss coin ...

(You could add another 50 days of "do nothing" or "do not wake sleeper" every time it's heads, if you want, but it makes no difference to the number of questions asked.)

We expect the experiment to last 150 days. On 100 of those days the correct answer is "tails" and on only 50 days the correct answer is tails.

If the sleeper answers Heads every time they are asked, they win 50/150 times.

If the sleeper answers Tails every time they are asked, they win 100/150 times.

Do you accept these calculations?

Yes I accept these calculations but like I’m explaining to Dale winning more doesn’t change the probability.

You should think of the coin toss as deciding which world she is entering. When she wakes up the chances of her being in the world where the coin landed heads or the world where the coin landed tails is 50/50. When you ask her what her credence of heads is she needs to look at the probability of her being in either world. The chances are 50/50.

 

[PeroK]

Yes I accept these calculations ...

Would the sleeper, when she is woken, be able to perform those calculations - each and every time?

 

[Dale]

I agree in the first bet she should choose tails

That is the credence. This is the number she is supposed to provide.

She is not asked whether she thinks she chose the better option but rather if she thinks she will win the first bet.

I understood your point. I think you are wrong, but my usual approach to such things is to just run a Monte Carlo simulation and see. I will do that later.

However, this second question is not the credence that she is being asked to calculate. So the result of the simulation will surprise one of us, but not change the discussion at all.

Please answer this question and do not avoid it again: Since you have proposed an alternate, let me do so as well, we will call this one the “Extreme Edition”:

Suppose that she is told in the beginning that if it is heads she will not be woken and interviewed on either Monday or Tuesday while if it is tails she will be woken and interviewed on Monday and Tuesday. What should be her credence for heads then? Why?

 

[PeroK]

Please answer this question and do not avoid it again: Since you have proposed an alternate, let me do so as well, we will call this one the “Extreme Edition”:

Suppose that she is told in the beginning that if it is heads she will not be woken and interviewed on either Monday or Tuesday while if it is tails she will be woken and interviewed on Monday and Tuesday. What should be her credence for heads then? Why?

To be fair, this puzzle does manage to introduce an experiment apparently without giving the amnesiac any information that she did not have initially. The only subtlety I can see is precisely why the halfer position is wrong. In this extreme version, the halfer position is clearly wrong.

The thirder position involves nothing but standard probability calculations (that the amnesiac can equally well carry out - given she knows the rules of the game, but has no knowldege of the stage the game has reached). That's why the thirder position is clearly correct.

Ultimately, the halfer position is an appealing intuitive short-cut that gives a different answer from the expected answer. The effort in this situation must be to establish why the intuitive short-cut is wrong. This happens to me quite often, where I have a quick and easy intuitive solution that gives the wrong answer. There's always a flaw somewhere.

 

[Moes]

That is the credence. This is the number she is supposed to provide.

I understood your point. I think you are wrong, but my usual approach to such things is to just run a Monte Carlo simulation and see. I will do that later.

However, this second question is not the credence that she is being asked to calculate. So the result of the simulation will surprise one of us, but not change the discussion at all.

Please answer this question and do not avoid it again: Since you have proposed an alternate, let me do so as well, we will call this one the “Extreme Edition”:

Suppose that she is told in the beginning that if it is heads she will not be woken and interviewed on either Monday or Tuesday while if it is tails she will be woken and interviewed on Monday and Tuesday. What should be her credence for heads then? Why?

In the Extreme addition you propose she gains new information by waking up. So her credence for heads is now 0.

I didn’t avoid the question I just gave a short answer. I don’t know if I can explain more than that. In your extreme addition she clearly gets new if information. She has woken up. What were the chances of that happening if the coin landed heads? 0.

In the original version I thought you already understood my argument that when she wakes up and asks herself this question the answer is not a 50% chance. Even if the coin landed heads there is a 100% chance that she will be woken up one day and that is all that happened.

I think we we should go back to the question I’m asking you, because I think there is no choice but for you to agree that the second bet will be 50/50. And according to how your calculating her credence you end up in a contradiction. When we talk about her credence we must mean also what her credence about her original credence is. You can’t believe something but at the same time not be sure if you believe what you believe. Maybe you can play around with the definition of belief, but then you are just confusing people about what your talking about.

 

[PeroK]

When we talk about her credence we must mean also what her credence about her original credence is.

Her original credence is $1/3$. That's the point. It doesn't change. You agreed with that when you accepted the calculation in post #58.

 

[Filip Larsen]

The thirder position involves nothing but standard probability calculations (that the amnesiac can equally well carry out - given she knows the rules of the game, but has no knowldege of the stage the game has reached). That's why the thirder position is clearly correct.

What if one model it from the sleepers perspective where she just experience the event of being awaken (A) without knowledge of days? With such a model of the events I get P(H|A)P(A) = P(A|H)P(H), which for P(A) = 1 and P(H) = 0.5 gives P(H|A) = 0.5. Now I did a proper calculation and the result is now 0.5. Is this model less valid than the original with A = Monday or Tuesday? This simpler model captures the halfers position, and while different why should that be less correct?

 

[PeroK]

What if one model it from the sleepers perspective where she just experience the event of being awaken (A) without knowledge of days? With such a model of the events I get P(H|A)P(A) = P(A|H)P(H), which for P(A) = 1 and P(H) = 0.5 gives P(H|A) = 0.5. Now I did a proper calculation and the result is now 0.5. Is this model less valid than the original with A = Monday or Tuesday? This simpler model captures the halfers position, and while different why should that be less correct?

I don't know how you got that answer.

 

[Moes]

Her original credence is $1/3$. That's the point. It doesn't change. You agreed with that when you accepted the calculation in post #58.

I’m not sure your following my argument with Dale. But yes, it shouldn’t change but according to how you calculate credence with betting it doesn’t make sense. In your own 150 day version she would come out winning the second bet 50% of the time either way she places her bet so her credence should be .5

 

[PeroK]

I’m not sure your following my argument with Dale. But yes, it shouldn’t change but according to how you calculate credence with betting it doesn’t make sense. In your own 150 day version she would come out winning the second bet 50% of the time either way she places her bet so her credence should be .5

In a previous post, you agreed with this:

If the sleeper answers Heads every time they are asked, they win 50/150 times.

If the sleeper answers Tails every time they are asked, they win 100/150 times.

Do you accept these calculations?

That's a credence of $1/3$ for Heads. A credence of $0.5$ would yield $75/150$ successes.

And, of course, she can work out all this advance and again every time she is woken. Nothing changes for her.

 

[Moes]

In a previous post, you agreed with this:That's a credence of $1/3$ for Heads. A credence of $0.5$ would yield $75/150$ successes.

I don’t believe you are following my argument with Dale. This is the question I asked him.

If after she is woken up, and puts her bet on tails, she is offered another bet that she is told will only be offered by this one awakening.
The bet is about whether she just won the previous bet. Are the chances not 50/50 on this bet?

I believe this leads to a contradiction in your line of reasoning

I think you should think through my case again because if you still argue there then I’m miss understanding your argument

The main difference I was making with the second bet was that it is only offered in one awakening whether the coin landed heads or tails.

I thought you agreed the probability of the coin toss remains 50/50. You were just arguing that since betting on tails comes out to her advantage she should for some reason believe in tails more.

By the second bet she is asked whether she thinks she won the first bet or not. The chances of her winning the first bet was 50/50. 50% chance she loses and 50% she wins double.

She is not asked whether she thinks she chose the better option but rather if she thinks she will win the first bet.

I can make a random bet where there is a 50% chance I lose 10 dollars and 50% chance I win 20. The other person may be stupid for agreeing to it but it won’t mean I have a better chance of winning the bet. The money won’t change the 50/50 chance.(I agree the circumstances are different here but for what I’m arguing that doesn’t matter)

So although I agree in the first bet she should choose tails in the second bet she gets no advantage if she chooses tails since she has no chance of winning double. She is only offered the bet once.

Where exactly are you arguing?

how are you answering my question?

 

[PeroK]

I don’t believe you are following my argument with Dale. This is the question I asked him.how are you answering my question?

I can't see the relevance of that question.

 

[Dale]

In your extreme addition she clearly gets new if information

So being awoken can indeed convey information. What is it about the extreme experiment that provides the information?

I think we we should go back to the question I’m asking you, because I think there is no choice but for you to agree that the second bet will be 50/50.

I don’t yet agree, but I will run a Monte Carlo simulation to find out. There is nothing to discuss until I do that (or you can do such a simulation and post the results).

However, that number is not the credence being requested. So although I want to find it out to see if my intuition is right (I am willing to correct my intuition if it disagrees with the math), it is not relevant to the credence.

When we talk about her credence we must mean also what her credence about her original credence is.

That is silly. The credence of the credence is not the credence. Math doesn’t work that way. Not all functions are the identity function. Particularly since she is specifically told that this second credence is to be calculated differently.

 

[PeroK]

So being awoken can indeed convey information.

That the game has begun!

 

[Moes]

So being awoken can indeed convey information. What is it about the extreme experiment that provides the information?

.

That the game has begun!

In the extreme experiment the new information is not just that the game has begun. It’s that she ’beat’ the chances of her not waking up. This is useful information. In the original experiment the new information is just that the game has begun. This is useless information. It was always known the game would begin.

That is silly. The credence of the credence is not the credence. Math doesn’t work that way. Not all functions are the identity function. Particularly since she is specifically told that this second credence is to be calculated differently.

This is not making sense in my language.

And anyways, if we need to pick one, it’s the second bet that properly calculates her overall credence.

 

[Moes]

Math doesn’t work that way.

Its not about how math works it’s about how your explaining what we call her belief.

 

[Office_Shredder]

I feel like the entire problem with this thread is that probability is intended to tell you the fraction of times a certain outcome will occur if you run a trial repeatedly, and Moes doesn't believe that.

 

[Dale]

It’s that she ’beat’ the chances of her not waking up. This is useful information.

Yes, she beat the chances of not waking up. In other words, the fact of waking can convey information about the coin toss.

In the original experiment the new information is just that the game has begun.

In the extreme case the information is completely unambiguous. However, information need not be completely unambiguous to be information.

Let’s consider an Almost Extreme example. They purchase a lottery ticket. If they win the 1 in 10 million prize then they will play the Original version and otherwise they play the Extreme version.

So now there is some ambiguity. What do you think happens to the credence of heads upon being awoken now? According to you, the credence in the Extreme case is 0 because she obtained information by waking. And according to you, the Credence in the Original case is 0.5 because there is no information by waking. So how about Almost Extreme? Does waking still provide information?

And anyways, if we need to pick one, it’s the second bet that properly calculates her overall credence.

Frankly, this is nonsense because she is asked to state her credence every time she is awoken and she is explicitly not asked your second bet every time she is awoken. There is no possible way that the two can be equivalent since your second bet doesn’t even produce a number in all of the cases that her credence does.

Anyway, if you wish to claim that such a silly procedure is the one described by the request for a credence then I want to see a professional scientific reference that supports this. I remind you of the rules of this forum which require all posts to be consistent with the literature and the long-standing tradition that such requests for references should be honored or the claim rescinded.

 

[Moes]

I feel like the entire problem with this thread is that probability is intended to tell you the fraction of times a certain outcome will occur if you run a trial repeatedly, and Moes doesn't believe that.

I think I could agree with that. I just think you need to know how to use this rule in the right way.

If the experiment was run 50 times in 25/50 of the experiments the coin should land heads and in 25/50 it should land tails. When she is woken up and asked what her credence is that the coin landed heads the question she is being asked is which kind of experiment does she think she is in. The chances of being in an experiment where the coin landed heads is 50%.

Just to be clear - the fraction of times the outcome of being in an experiment where the coin landed heads is 25/50.

 

[Moes]

Yes, she beat the chances of not waking up. In other words, the fact of waking can convey information about the coin toss.

In the extreme case the information is completely unambiguous. However, information need not be completely unambiguous to be information.

Let’s consider an Almost Extreme example. They purchase a lottery ticket. If they win the 1 in 10 million prize then they will play the Original version and otherwise they play the Extreme version.

So now there is some ambiguity. What do you think happens to the credence of heads upon being awoken now? According to you, the credence in the Extreme case is 0 because she obtained information by waking. And according to you, the Credence in the Original case is 0.5 because there is no information by waking. So how about Almost Extreme? Does waking still provide information?

Waking up provides her with information that she would’ve had to have won the lottery for it to be possible the coin landed heads. This is very unlikely so she should assume it landed tails. I think it should be something like a 1 in 20 million chance that the coin landed heads. I don’t think waking up provides any other useful information.

I think it may be better if we leave this point, we are getting closer to understanding our argument with your next point:

“I agree. The chances of being in an experiment where the coin landed heads is 50%. That is not the credence.”Would you also say she should believe the coin probably landed tails despite the fact that the chances it landed tails is only 50%?

 

[Dale]

This is very unlikely so she should assume it landed tails. I think it should be something like a 1 in 20 million chance that the coin landed heads.

OK, so there is, in fact, a continuum of credences. A little ambiguity does not completely destroy the information.

So what happens as we change Almost Extreme by using the 1 in 1000 prize, or the 1 in 10 prize, or a guaranteed prize? Hopefully you agree that the credences gradually change from the Extreme credence to the Original credence.

So, you agreed that waking in the Extreme case and in the Almost Extreme case conveyed information. You also stated that waking provided no information in the Original case. But the Original case is just the Almost Extreme case with a guaranteed lottery prize. So exactly where on the continuum from 1 in 10 million to guaranteed Sid waking go from providing some information to providing no information?

 

[Dale]

Would you also say she should believe the coin probably landed tails despite the fact that the chances it landed tails is only 50%?

When she is awoken and interviewed, yes. Then her credence is 1/3 as those are the odds she would bet.

 

[Moes]

OK, so there is, in fact, a continuum of credences. A little ambiguity does not completely destroy the information.

So what happens as we change Almost Extreme by using the 1 in 1000 prize, or the 1 in 10 prize, or a guaranteed prize? Hopefully you agree that the credences gradually change from the Extreme credence to the Original credence.

So, you agreed that waking in the Extreme case and in the Almost Extreme case conveyed information. You also stated that waking provided no information in the Original case. But the Original case is just the Almost Extreme case with a guaranteed lottery prize. So exactly where on the continuum from 1 in 10 million to guaranteed Sid waking go from providing some information to providing no information?

I feel like you are just playing with words here.

There are 2 parts to the Almost extreme version. There is the chance she will enter the original version and a chance she will enter the extreme version. Waking up only provides information provided that she assumes she entered the extreme version. At the end of the continuum where the chance of her entering the extreme version is eliminated is where waking up can obviously not provide her with information about an experiment that doesn’t even have a chance of existing.

 

[Moes]

When she is awoken and interviewed, yes. Then her credence is 1/3 as those are the odds she would bet.

Just one more confirmation - when you say “When she is awoken and interviewed” you still agree the chances of the coin toss still didn’t change. True?

 

[PeroK]

Just one more confirmation - when you say “When she is awoken and interviewed” you still agree the chances of the coin toss still didn’t change. True?

Are there any circumstances where the credence that a coin landed Heads is not one of $0, 0.5$ or $1$?

After you toss a fair coin, the credence that it is Heads must be $0.5$, always. And, given that the coin itself never changes its position, surely it must always be $0.5$ - until you look at it and it becomes Heads or Tails with certainty?

Is $1/3$ ever a valid credence for a fair coin toss?

 

[PeroK]

I feel like the entire problem with this thread is that probability is intended to tell you the fraction of times a certain outcome will occur if you run a trial repeatedly, and Moes doesn't believe that.

Or, the point we reached in the previous thread a few years ago, was that the probability that a coin is Heads can only be one of $0, 0.5$ or $1$ and the thirder position is absurd from that point of view as well.

There's a certain truth in this, if you subtlely keep your question focused on the original coin toss and the credence immediately thereafter, and don't allow the question to change to "what is your credence now that someone has looked at the coin and acted on it".

Changing your credence about a coin that has not been touched may be a psychological barrier and those of us with knowledge of modern statistics and probability theory perhaps underestimate how difficult a concept that may be.

 

[PeroK]

When she is awoken and interviewed, yes. Then her credence is 1/3 as those are the odds she would bet.

Here's hopefully a further insight into the information that the sleeper has.

When she is awoken the information she has is this: that someone has looked at the coin and acted on what they saw.

Or, in fact, more subtlely: that there is a possibility that someone has looked at the coin and acted on what they saw.

Her credence remains $1/2$ only so long as she is certain that no one has looked at the coin; or, if they have, she is certain that they have not yet acted on what they saw.

This is what's different when she is awoken. The certainty that no one has acted on what the coin shows has gone. That loss of certainty about this is the new information.