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

[PeroK]

So you and Dale are saying, that credence is to be understood as an equivalent question to the sleeping beauty along the line "given all you know about this experiment are you willing to bet that the coin came up on tails" i.e. the question she is asked each day she is awoken is for her to be understood to have consequences even if she herself cannot remember them?

Yes, and I would give a full analysis here.

We have X, who runs the experiment and S, who is the sleeper.

At the start of the experiment the coin is tossed, but no one looks at it. The credences that it is Heads should be:

X = 1/2
S = 1/2

Now, we run the experiment:

Suppose the coin is Heads. This means S gets woken only once. The credences are:

X = 1
S = 1/3

Suppose the coin is Tails. This means S gets woken twice:

First time:

X = 0
S = 1/3

Second time:

X = 0
S = 1/3

The logic is that S cannot distinguish between these three scenarios, which are all equally likely. Note that S can give these answers in advance (before they are woken). There is no new information. The information given at the outset was enough to predict that their credence would always be $1/3$.

Note that S is "correct" in the sense that they give Heads only a 1/3 probability each time and it is Heads 1/3 of the time. They are not using any information about the state of the game, only information about the rules of the game (which was available at the outset). So, no new information.

A sleeper who gives Heads a probability of 1/2 each time is wrong in the sense that they are choosing to ignore the information that they have about the rules of the game.

 

[Moes]

The argument for 50-50 is the one that you have given: it must be 50-50 and no rational person would disagree.

The argument for a third is to do the calculations and find that the answer is one third. There are similar problems where the answer is one third by the same calculations. The 50-50 position, however, is that you are not allowed to do calculations or use mathematics in this specific problem.

This is based on the dubious logic that the sleeping beauty is a princess, not a mathematician, and not capable of mathemetical calculations.

We had a long thread on this a few years ago that turned rather nasty and almost led to my leaving PF.

I wouldn’t say you are not allowed to do calculations just because she is not a mathematician rather my claim is the calculations just don’t apply here or your just using the wrong ones. The answer to the problem wouldn’t change whether she was a mathematician or not.

I am trying to argue that the game version of the experiment is a different experiment than the original one (as it is described in the Wikipedia link at least). There is no mentioning of betting or sums of wins/losses in the original version, and I argue that just because its a winning strategy for her to answer tails in some game version of the experiment it does not by itself imply that the answer the question "What is your credence now for the proposition that the coin landed heads?" has the best answer of 1/3. I here stress that I assume that this is the point of the game-argument is trying to make, namely to say that since there is a winning strategy saying tails then tails must be more likely than heads. If that was not yours or Dales intention then I may have misunderstood what argument you are trying to make.

Going back to the original experiment, the key point for me is that she is asked the same question with no change in her knowledge from even before the experiments starts (as we all seem to agree on). So why would she believe P(heads) = 1/3? If I toss a fair coin and ask you to estimate P(head) why would it matter I ask you once more? I mean, you would not say P(head) = 1/2 the first time I ask and then suddenly say P(head) = 1/3 just because I ask again (even if you knew in advance that I would potentially ask twice), so what is the difference in knowledge between you and the sleeping beauty that should make her claim P(head) = 1/3?

This is basically my claim. I still don’t see how anyone can argue with this. Using the practical betting scenario: Say she was betting 10 dollars on correctly guessing heads or tails. If she knew she was going to be put into this bet every time she woke up then I understand she should choose tails. This is because choosing tails gives her a 50/50 chance of winning 20 dollars or losing just 10. Whereas if she chooses heads it’s vise versa. 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.)
(What if she is asked to put one bet on whether she thinks she will gain or lose money with this whole experiment? The chances are surely 50/50 here.)

When she is in the experiment she only knows of being ask into this bet once. 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. So In this case there is no advantage for her to choose heads or tails when she is woken the chances of her winning or losing the bet are simply 50/50.

I don’t see anyone getting around this. I’m only hearing arguments about what exactly is the question at hand. And when the word “credence” is used I don’t see any room for argument her credence is simply 50/50 she could not be any more sure that the coin landed either heads or tails.

 

[Filip Larsen]

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.

 

[PeroK]

This is basically my claim. I still don’t see how anyone can argue with this. Using the practical betting scenario: Say she was betting 10 dollars on correctly guessing heads or tails. If she knew she was going to be put into this bet every time she woke up then I understand she should choose tails. This is because choosing tails gives her a 50/50 chance of winning 20 dollars or losing just 10. Whereas if she chooses heads it’s vise versa. 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.)

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

 

[PeroK]

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).

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

I'm happy to be a thirder and retain a working probability theory. And, I'll let the halfers sift through the wreckage of what's left of their probability theory.

 

[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 $1/3$.)