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

[Moes]

In order to understand the consequences of how the word "credence" in the question affects the sleepers analysis, please consider a variation of the experiment where all is as original except that no matter what the coin lands on, the experimenter will only ask the sleeper the question once. If the coin lands on tails the experimenter will choose at random with day (Monday or Tuesday) he will ask and the other day he will just say "sorry, no question today". The sleeper will know this rule in advance, but when awaken will of course still not remember if he has been awaken or asked before (which means the rules of this variant could equivalently be that the sleeper is only awoken once no matter how the coin lands). According how you define credence, I would think the sleeper should now believe that both heads and tails are equally likely because if he were to bet on either he can expect to come out of the experiment at zero win on average. Is this correct?

If correct I think this way of considering credence to be a measure of probability is "broken" for this experiment (which is no doubt formulated in this way to bring out such conflict). At least it sounds very paradoxical to me that by promising not to repeat a question later that, if asked, is guaranteed to yield same answer, you can somehow affect that answer.

Sorry I am not understanding your question can you please
explain the variation of the experiment at little clearer and explain what you think the answer should actually be and what it should be according to my way of considering credence?

 

[Moes]

Winning bets is precisely what probabilities are about. That's fundamental.

Can you explain more clearly where and how your arguing with what I was saying

winning bets is only precisely what probability is about when there is a direct correlation with the bet won and the probability your talking about. I don't see that here.

 

[PeroK]

winning bets is only precisely what probability is about when there is a direct correlation with the bet won and the probability your talking about. I don't see that here.

Any probability can be directly turned into a bet. The bet simply illustrates the probability. To take my example: it's not possible to argue that the probability of a die coming up six is $1/2$ and know that it only comes up $1/6$ of the time. That would imply an inconsistency between probabilities and outcomes,

More generally, you can look at credence (see the above link) as optimising your winning chances given the information you have. In this case, if the sleeper ignores what she knows about the game, and calculates $1/2$, then that is sub-optimal. But, if she uses the information she has about the game, then she can calculate the probability of $1/3$, which turns out to be correct. Certainly better than the $1/2$ at least.

This is the problem with the halfer argument. You can convince yourself that the probability is $1/2$, but at the same time you have to admit that someone who uses the full information at their disposal does better (probabilistically). You can only accept both of these if you demand that this particular problem breaks probability theory.

I prefer to take the view that probability theory can handle the sleeping beauty problem and is not broken by it.

 

[Moes]

Information does not require that you be able to definitely determine a hypothesis. It only requires that the information be more likely under one hypothesis than under the other. In this case the fact that she is awoken is twice as likely under one hypothesis than under the other. So the fact that she is awoken is indeed information.

This is exactly what I disagree with. its hard for me to explain but I tried in my original message, see if you can understand what I was arguing there.

Does anyone understand how Dale could be wrong that she gets any new information? If yes please explain clearly why.

 

[Dale]

its hard for me to explain but I tried in my original message, see if you can understand what I was arguing there.

I understand your argument, I just disagree with it. Using the standard mathematical treatment of these concepts it is easy enough to show why I disagree with it. Your message is much much less convincing than the actual math.

 

[PeroK]

Your message is much much less convincing than the actual math.

This is why I believe the halfer argument is fundamentally an intuitive argument that appeals to rationality, but dare not do any calculations. And, the thirder argument is to eschew the intuition and do the math(s).

 

[Filip Larsen]

Sorry I am not understanding your question can you please

My reply was to PeroK, sorry for the confusion.

That's a good point. Either the thirders are right; or, the halfers are right and probability theory is broken

But you elegantly avoided my question I made in an attempt try understand how you would measure credence in a slightly different situation, so please allow me to ask again:

If the experiment is varied so the sleeper is only awaken once no matter what face the coin lands on (and the sleeper knows this), is it then correct to say that the sleeper should have credence that tails and heads have equal probability, i.e. if the sleeper bet in this situation the expected average win will be zero?

Please note, I am not trying to refute any official understanding of credence is, I am trying to understand how it differs from my understanding of probabilities in such constant knowledge and memory-less situations.

 

[PeroK]

If the experiment is varied so the sleeper is only awaken once no matter what face the coin lands on (and the sleeper knows this), is it then correct to say that the sleeper should have credence that tails and heads have equal probability, i.e. if the sleeper bet in this situation the expected average win will be zero?

Yes, if that's all the information you have.

 

[PeroK]

memory-less situations.

Just to be clear, the sleeper needs to retain knowledge of the rules of the game. Without that she has no information to work on.

 

[Filip Larsen]

Just to be clear, the sleeper needs to retain knowledge of the rules of the game. Without that she has no information to work on.

Yes, I agree on that. The lack of memory refers to the sleeper having no knowledge if she is already been awoken or not for each run of the experiment. As far as I can tell that is how "lack of memory" has been understood by everyone in this thread, at least in context of the original experiment.

 

[Dale]

the calculations just don’t apply here

That is not an acceptable position. You are asking her to produce a numerical value for credence. To say that calculations don’t apply for a request to produce a credence is folly. If they don’t apply then there is certainly no justification for the “halfer” position either.

or your just using the wrong ones.

That is an acceptable position, but you need to back it up with the correct calculations.

If she knew she was going to be put into this bet every time she woke up then I understand she should choose tails

She does know. She is explicitly told beforehand that she will be asked to state her credence every time she is awoken and interviewed.

However this doesn’t show you anything at all about what her credence should be which is the question at hand. (More money doesn’t up the probability of heads or tails.)

Yes it does. That is how credence is operationally defined.

So to properly calculate what her credence should be, is to ask if she is told she will only be placing a bet on one awakening whether the coin landed heads or tails.

I don’t see how that can be justified. She is told explicitly that in the case of tails she will be asked to state two credences. How can you justify treating two credences as one bet. Have you any scientific reference that supports this approach?

This is my issue with the “halfer” position. Yes, you can certainly come up with wagers that would produce a credence of 0.5. But such wagers are not related to the credences in the scenario by any means that I have ever seen in the literature. I have never seen any source take two credences and treat them as the odds on one wager. Have you?

 

[Moes]

Your message is much much less convincing than the actual math.

This is why I believe the halfer argument is fundamentally an intuitive argument that appeals to rationality, but dare not do any calculations. And, the thirder argument is to eschew the intuition and do the math(s).

It sounds like your saying to ignore pure logical reasoning and go with mathematic calculation. I don’t understand how you could do that. It’s not just one of our intuitions that are many times wrong. Logically I don’t see any way around the halfer argument. I am not much of a mathematician so I can’t really argue about the calculation.

Are you really saying she can be more sure the coin landed tails than heads?
Just because guessing tails gives her a better advantage of having the chance to be right twice that doesn’t mean she can be more convinced that it actually landed tails.

is anyone arguing with this or are we just arguing linguistics here?

credence is defined as belief in or acceptance that something is true.

 

[Moes]

If she knew she was going to be put into this bet every time she woke up then I understand she should choose tails

“She does know. She is explicitly told beforehand that she will be asked to state her credence every time she is awoken and interviewed.”i was trying to say I agree she should “choose tails” but that shouldn’t mean she should believe that it was probably tails. Therefore using this betting scenario just doesn’t help us calculate the probability here. I therefore suggested a different betting scenario which I believe better calculates the probability.

 

[Office_Shredder]

Let's suppose she's not woken up every time they flip heads.

In fact, we're going to do the following. We're only going to wake her up at most once, but with the following rules. If the coin lands on heads, we flip it again, and wake her up if it lands heads a second time. If it lands tails, then we wake her up once, without flipping the second time.

What is the credence that the coin came up tails when she gets woken up?

This is the same problem as the original problem, but we just wake her up half as often. 0.5 times instead of once for heads, once instead of twice for tails.

In particular, playing this game four times, twice if lands on tails, one it goes heads tails, once it goes heads heads, is the same as playing the original game twice.

 

[Dale]

i was trying to say I agree she should “choose tails” but that shouldn’t mean she should believe that it was probably tails.

What do you mean “it was probably tails”. I suspect that you are incorrectly thinking that when Sleeping Beauty is asked to state her credence that she is being asked to state the probability that a flip on a fair coin produces tails. That is not what she is being asked.

Here is a nice description of credence:

https://acritch.com/credence/

the 90% in “I’m 90% sure” is called your credence, and the phrase

“I’m 90% sure that Joe is at the party”
is defined to mean roughly that

“I’d rather bet that Joe is at the party than bet on an 89%-biased roulette wheel, and I’d rather bet on a 91%-biased roulette wheel than bet that Joe is at the party.”

When she is woken and interviewed she is asked her credence that it was heads. As described above, this means the bias level at which she becomes indifferent to a bet on heads or a bet on the biased roulette wheel.

This credence is different than simply the probability that a fair coin flip turns up heads because the setup is more involved than simply a coin flip.

 

[Moes]

What do you mean “it was probably tails”. I suspect that you are incorrectly thinking that when Sleeping Beauty is asked to state her credence that she is being asked to state the probability that a flip on a fair coin produces tails. That is not what she is being asked.

Here is a nice description of credence:

https://acritch.com/credence/
When she is woken and interviewed she is asked her credence that it was heads. As described above, this means the bias level at which she becomes indifferent to a bet on heads or a bet on the biased roulette wheel.

This credence is different than simply the probability that a fair coin flip turns up heads because the setup is more involved than simply a coin flip.

I understand this. However in this case she would only bet on tails because it gives her a 50% chance of losing money and 50% chance of winning double money. Whereas if she places a bet on heads she has a 50% chance of winning money and 50% chance of losing double. So to bet on tails is just a better bet to make. But the probability of heads or tails stays 50/50.

Am I not making sense here?

 

[Dale]

So to bet on tails is just a better bet to make. But the probability of heads or tails stays 50/50.

Yes, and yes. The coin is fair so the probability of heads from a single flip indeed stays 50/50, but the rational bet is not 50/50 so neither is the credence.

Am I not making sense here?

Yes, but you seem to not want to take the clear next step. Your own reasoning shows that the rational bet is not 50/50.

 

[Moes]

Let's suppose she's not woken up every time they flip heads.

In fact, we're going to do the following. We're only going to wake her up at most once, but with the following rules. If the coin lands on heads, we flip it again, and wake her up if it lands heads a second time. If it lands tails, then we wake her up once, without flipping the second time.

What is the credence that the coin came up tails when she gets woken up?

This is the same problem as the original problem, but we just wake her up half as often. 0.5 times instead of once for heads, once instead of twice for tails.

In particular, playing this game four times, twice if lands on tails, one it goes heads tails, once it goes heads heads, is the same as playing the original game twice

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.

 

[Moes]

Yes, and yes. The coin is fair so the probability of heads from a single flip indeed stays 50/50, but the rational bet is not 50/50 so neither is the credence.

Yes, but you seem to not want to take the clear next step. Your own reasoning shows that the rational bet is not 50/50.

Maybe what your calling “the rational bet” is not 50/50 but do you agree her belief in tails shouldn’t be any stronger than it is of heads?

 

[Dale]

Maybe what your calling “the rational bet” is not 50/50 but do you agree her belief in tails shouldn’t be any stronger than it is of heads?

Since Sleeping Beauty is described as rational then the rational bet and her belief are one and the same.

 

[Moes]

Since Sleeping Beauty is described as rational then the rational bet and her belief are one and the same.

In this case i don’t understand why that’s true.

 

[Moes]

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

 

[PeroK]

I understand this. However in this case she would only bet on tails because it gives her a 50% chance of losing money and 50% chance of winning double money. Whereas if she places a bet on heads she has a 50% chance of winning money and 50% chance of losing double. So to bet on tails is just a better bet to make. But the probability of heads or tails stays 50/50.

Am I not making sense here?

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?

 

[Dale]

In this case i don’t understand why that’s true.

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.

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

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?

 

[PeroK]

I want to propose an analysis of this problem that exposes the information paradox.

First, at the outset we ask everyone involved what their credence will be on Heads on a randomly selected future day.

(Note: it not the credence that coin is Heads now (both should answer $1/2$ to that), but the credence on a future random day.)

The experimenter, having no information about the toss or specific day, answers $1/3$ (using the standard analysis in post #58) above.

The sleeper, likewise, answers $1/3$.

Now, on the first day, let's assume that the sleeper is woken before the coin is looked at.

The sleeper does not know what day it is and has no new information. She must stay with her a priori answer of $1/3$.

The experimenter, however, has the information that this is the first day. He has new information and changes his credence to $1/2$.

After looking at the coin, the experimenter further changes his credence to $0$ or $1$, as appropriate.

This is where the information paradox lies. The problem is presented in such a way as to suggest that the sleeper changes her answer in the absence of new information. But, in fact, the sleeper never changes her answer. It must remain $1/3$ throughout. The experimenter, however, is the one who changes their credence based on new information throughout the experiment.

(Note that if the coin is tails, then on the second day the experimenter has a credence of $0$ for Heads; whereas, the sleeper still has no new information and remains with $1/3$.)

 

[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?