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

[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.

 

[Office_Shredder]

[

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

But this is clearly not the question, and you've even acknowledged this is in other examples.

Let's try another experiment. We flip a coin. If it's heads, it's over. If it's tails, I ask you if it's heads or tails.

If we run this experiment 50 times, there will be 25 experiments with heads, and 25 with tails, but the right answer to my question is still tails.

Let's strip out the coin. There are two colors, red and blue. Over the course of 100 days, I'm going to ask you 75 times whether the color is red, or the color is blue. I'm not going to tell you how I pick which color is right, but I will tell you right now that the right answer is blue 50 times, and red 25 times.

If I ask you the question, what is your credence that the color is blue? Did it matter *how* I generated which days it would be blue and which days it would be red?

 

[Dale]

I feel like you are just playing with words here.

It is actually playing with math here, but without formulas. I have made a scenario which varies smoothly from two scenarios where you agree that being woken and interviewed conveys information to a scenario where you claim it provides no information. The obvious question is where on that smooth continuum does it suddenly jump from providing information to not providing information.

Anyway, perhaps the discussion about information is tangential. If you are uninterested in discussing it then just think about it.

I guess the real issue is more about the concept of credence itself. You agree that the rational bet probability is 1/3, but insist that the credence is 1/2 anyway. I don’t know how to overcome that other than to point out again the definition of credence.

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?

I confirm that a fair coin toss is always a probability of 0.5. Mathematically in this problem $P(heads)=0.5$ always since we are explicitly assuming a fair coin.

What she is asked to provide, however, is her credence on $P(heads|awoken)$ which is equal to the rational bet she would make. The fact that $P(heads|awoken)=1/3$ in no way alters the fact that $P(heads)=1/2$.

The issue is that you agree that the rational bet is $1/3$ but refuse to associate that with the credence as you should. I suspect that it may be that you think she is being asked to state $P(heads)$ to which the correct answer is indeed $1/2$. But she is not being asked that.

 

[PeroK]

I guess the real issue is more about the concept of credence itself. You agree that the rational bet probability is 1/3, but insist that the credence is 1/2 anyway. I don’t know how to overcome that other than to point out again the definition of credence.

The truth of the matter is that it is almost impossible to change anyone's mind on these or similar problems. We were nearly there with @Filip Larsen, but it's not going to happen. Once someone has decided that the answer is $1/2$ there is no power on this Earth that will change their mind.

 

[Dale]

The truth of the matter is that it is almost impossible to change anyone's mind on these or similar problems.

This one seems even more challenging than the Monty Hall problem. I wish people would just learn to do Monte Carlo simulations.

 

[Moes]

It is actually playing with math here, but without formulas. I have made a scenario which varies smoothly from two scenarios where you agree that being woken and interviewed conveys information to a scenario where you claim it provides no information. The obvious question is where on that smooth continuum does it suddenly jump from providing information to not providing information.

I thought I already answered this question. Only when the chance of her being in the extreme version is completely eliminated, does she not gain any information about that extreme version.But let’s get to our real argument. Let me try to explain where your going wrong.

I think the more obvious something is the harder it gets to explain.

Please don’t read this just looking for what you can argue on. Try to understand my view.

I think there is a problem with the way your defining credence. Using my example of the second bet, it comes out you are saying it’s possible she can believe something but then only be 50% sure her belief is correct. Which means she doesn’t believe what she believes. This to me is just not English.

Math should not change your definition of a word. Now let me explain where I think you are going wrong.

Her bet is not a REASON she should believe something. It is merely a test that can give us a SIGN to what she believes. Or if you want to start from belief you can say her belief causes her to bet a certain way. A belief and a bet are not identical. If you are wondering why mathematicians will define credence as in regards to a bet, this just a way to give a measurement to her level of belief.

Now, in this experiment the way you are adding a bet to the situation is flawed. For this bet there is another outside reason besides her belief that is causing her to bet in a certain way. The reason is that the way she places her bet changes the actual conditions of the bet.

This is why I think my case of the second bet is a more accurate way to calculate her credence. If you consider the second bet as a separate question then I think it is still possible to place the first bet correctly. Even in the first bet you can tell her you will only be offering this bet to her once. The amount of times she knows she will be asked the question about her credence shouldn’t change her belief. All asking her the question just once will do is not let reasons outside her belief make her decision.

If you don’t agree with this then it means you just have a different definition of credence. The common dictionary doesn't define credence your way. So when you say her credence is 1/3 you are just misleading many people.

The truth of the matter is that it is almost impossible to change anyone's mind on these or similar problems. We were nearly there with @Filip Larsen, but it's not going to happen. Once someone has decided that the answer is $1/2$ there is no power on this Earth that will change their mind.

Obviously what I think here is just the exact opposite. I just hope we can both stay open minded.

 

[Office_Shredder]

Let's strip out the coin. There are two colors, red and blue. Over the course of 100 days, I'm going to ask you 75 times whether the color is red, or the color is blue. I'm not going to tell you how I pick which color is right, but I will tell you right now that the right answer is blue 50 times, and red 25 times.

If I ask you the question, what is your credence that the color is blue? Did it matter *how* I generated which days it would be blue and which days it would be red?

 

[Filip Larsen]

The truth of the matter is that it is almost impossible to change anyone's mind on these or similar problems. We were nearly there with @Filip Larsen, but it's not going to happen.

I think I have left the discussion with reasonable understanding of the two positions and specifically the model and calculation that lies behind the result of 1/3. Also, I have been trying without success to find a real world problem that would map one-to-one to the sleeping beauty problem and without such I, as an engineer mostly interested in practical applications of Baysian probabilities, probably considering the sleeping beauty problem to be a fairly narrow and contrived problem of mostly theoretical interest (narrow because most variations of the experiment seems to yield 1/2 and contrived because it apparently has no real world equivalent).

So in short, I agree that for the given problem and the given event model the result cannot really sensibly be other than 1/3. But you are right in the sense that I use this result more to conclude that the event model the sleeper would use to calculuate her credence must be inappropriate.

 

[Moes]

[
But this is clearly not the question, and you've even acknowledged this is in other examples.

Let's try another experiment. We flip a coin. If it's heads, it's over. If it's tails, I ask you if it's heads or tails.

If we run this experiment 50 times, there will be 25 experiments with heads, and 25 with tails, but the right answer to my question is still tails.

Let's strip out the coin. There are two colors, red and blue. Over the course of 100 days, I'm going to ask you 75 times whether the color is red, or the color is blue. I'm not going to tell you how I pick which color is right, but I will tell you right now that the right answer is blue 50 times, and red 25 times.

If I ask you the question, what is your credence that the color is blue? Did it matter *how* I generated which days it would be blue and which days it would be red?

I think I finally understand your question here.

There is something wrong with the way your putting it. I think we can understand this better if we change the way the problem is set up. Let’s say the experiment is like this:
We flip a coin. If it lands heads sleeping beauty is woken up just once on Monday, at any random time. If it lands tails she is woken up twice on Monday at any two random times. I don’t think anything changed here.

Now, when she wakes up, the chances of it being a Monday where she would be woken up once or a Monday where she would be woken up twice is 50/50. What this means to me is that her credence that the coin landed heads is 1/2.

I’m not seeing how you could claim this calculation is wrong.

What your arguing is that since only 1/3 of the times she wakes up, it is a Monday where the coin landed heads, so the chances of this awakening being on a Monday where the coin landed heads is only 1/3. So this should be her credence.

I think the problem with this way of calculating is you are taking the scenario where the coin landed heads and the scenario where it landed tails and making it as if both possibilities actually happened( meaning as if she can think of herself as being in two worlds at once. Since she is in the experiment where she could only be in one of these worlds I don’t think she could think in this way ). You are doing this by thinking what would be if this experiment was repeated many times. I think you are wrong for doing this. She knows it can only be one type of Monday. Either it’s a Monday where she will be woken up once, meaning the coin landed heads, or its a Monday where she will be woken up twice , meaning the coin landed tails. This should mean her credence of heads should be 1/2.

I realize I am not being so clear here , I don’t really know how to explain this too well. I will try to think about it more.

In order for you to prove your view, I need a clear explanation exactly why my logic is wrong here.

I would like to point out that it seems that you and PeroK may not be agreeing with Dale about exactly why you think her credence is 1/3. Do you fully agree with him?

 

[Filip Larsen]

I think the problem with this way of calculating is you are taking the scenario where the coin landed heads and the scenario where it landed tails and making it as if both possibilities actually happened

This is where a "rephrasing" using words like bet and expected total win comes in handy as it at once tells you that it the full flow of the experiment that is consider and that it is the expected value that counts, hence "averaged over multiple experiments" where, for instance, half the times you expect heads and the other half tails. You could also say that the Bayesian conditional probability answers a different question than you think when hearing about the question asked to the sleeper.

 

[PeroK]

I would like to point out that it seems that you and PeroK may not be agreeing with Dale about exactly why you think her credence is 1/3. Do you fully agree with him?

We are all in agreement that the answer of $1/3$ is clearly correct. The subtlety in this problem is precisely why the argument for $1/2$ fails. And, generally, if an argument is flawed then there may be various ways of exposing the flaw. You can generally disprove things in a number of ways.

I've certainty seen more clearly in this thread why the halfer argument is wrong - and especially the fallacy that there is "no new information". That doesn't invalidate alternative analysis of why the halfer argument is wrong.

 

[Moes]

We are all in agreement that the answer of $1/3$ is clearly correct. The subtlety in this problem is precisely why the argument for $1/2$ fails. And, generally, if an argument is flawed then there may be various ways of exposing the flaw. You can generally disprove things in a number of ways.

I've certainty seen more clearly in this thread why the halfer argument is wrong - and especially the fallacy that there is "no new information". That doesn't invalidate alternative analysis of why the halfer argument is wrong.

Dale agreed to the following statement:

When sleeping beauty is woken up, the chances the coin landed heads is still only 50%.

Do you agree with this? You seem to have agreed with Office shredder. Which from what I understood disagreed with that.

 

[PeroK]

Now, when she wakes up, the chances of it being a Monday where she would be woken up once or a Monday where she would be woken up twice is 50/50. What this means to me is that her credence that the coin landed heads is 1/2.

I’m not seeing how you could claim this calculation is wrong.

I don't see how you could possibly claim that calculation is correct. It's flagrantly wrong.

You're saying that even if event A happens twice as often event B, both events are equally likely!

 

[PeroK]

Dale agreed to the following statement:

When sleeping beauty is woken up, the chances the coin landed heads is still only 50%.

Do you agree with this? You seem to have agreed with Office shredder. Which from what I understood disagreed with that.

When she is woken up the first time, it's 50%, but when she'd woken up the second time it's 0%.

 

[Moes]

I don't see how you could possibly claim that calculation is correct. It's flagrantly wrong.

You're saying that even if event A happens twice as often event B, both events are equally likely!

The event that the coin lands heads (which means it’s a Monday where she would be woken up only once) happens the same often as the event that the coin lands tails

 

[Moes]

When she is woken up the first time, it's 50%, but when she'd woken up the second time it's 0%.

Dale didn’t agree with that.

I don’t even understand what that means. She doesn’t know which day it is for there to be a first and second day.

I think Dale was pretty clear that even from sleeping beauty’s point of view the chances remain 50/50.

His claim was only that her credence should follow the way she would bet.