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

[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.

 

[PeroK]

Dale didn’t agree with that.

You need to stop playing both sides against the middle here. If @Dale agrees with you, let him say so himself.

 

[PeroK]

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

Are you assuming that she always thinks it's Monday?

 

[Moes]

You need to stop playing both sides against the middle here. If @Dale agrees with you, let him say so himself.

Sorry I guess your right and we just need to wait for Dale to reconfirm. But this is what I understood from his statement here.

.

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.

 

[Moes]

Are you assuming that she always thinks it's Monday?

I was talking about in my own version of the problem , sorry for the confusion.

. 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.

Your comment that I was replying to was also in this version

 

[PeroK]

I was talking about in my own version of the problem , sorry for the confusion.

Your comment that I was replying to was also in this version

Okay, are you assuming she always thinks it's the first time she's been woken up?

 

[Moes]

Okay, are you assuming she always thinks it's the first time she's been woken up?

No

 

[PeroK]

No

Well, that makes no sense. If she knows it might be the second time, then that is information that she can use. She has three equally likely events: H, T1, T2. That's a $1/3$ credence on Heads and $2/3$ on Tails.

 

[Moes]

Well, that makes no sense. If she knows it might be the second time, then that is information that she can use. She has three equally likely events: H, T1, T2. That's a $1/3$ credence on Heads and $2/3$ on Tails.

I don’t see how the 3 events are equally likely. There is a 50% chance the coin will land heads and she will wake up (H). And there is a 50% chance the coin will land tails. If it landed tails there is a 50% chance of T1 and 50% chance of T2 which makes T1 and T2 a 25% chance each.

 

[PeroK]

I don’t see how the 3 events are equally likely. There is a 50% chance the coin will land heads and she will wake up (H). And there is a 50% chance the coin will land tails. If it landed tails there is a 50% chance of T1 and 50% chance of T2 which makes T1 and T2 a 25% chance each.

That's elementary. If we get a Tail, then both events T1 and T2 take place. They both have a 50% probability of taking place.

 

[Moes]

That's elementary. If we get a Tail, then both events T1 and T2 take place. They both have a 50% probability of taking place.

We are not talking about the probability of them taking place. We are talking about the probability of her being up (presently) in one of those days.

 

[PeroK]

We are not talking about the probability of them taking place. We are talking about the probability of her being up (presently) in one of those days.

That makes no sense, not least because we are always talking about the same day.

The problem is that you have fundamentally misunderstood basic probability theory. This has nothing to do with the Sleeping Beauty puzzle.

You're calculating your own "probabilities" based on your own rules, that are different from what other people will calculate. We'd probably disagree about most things related to probability.

You say T1 has a probability of 25% and I say it has a probability of 50%. That's a fundamental difference that no discussion can ever resolve.

All I can say is that IF we start betting based on our different probability theories, then I'll win (on average). This is because the probabilities I calculate are based on relative frequencies and may be the basis for betting. Whereas, the probabilities you calculate do not relate to the relative frequency with which events take place and, therefore, cannot be the basis of betting without losing money (on average).

 

[Moes]

That makes no sense, not least because we are always talking about the same day.

The problem is that you have fundamentally misunderstood basic probability theory. This has nothing to do with the Sleeping Beauty puzzle.

You're calculating your own "probabilities" based on your own rules, that are different from what other people will calculate. We'd probably disagree about most things related to probability.

You say T1 has a probability of 25% and I say it has a probability of 50%. That's a fundamental difference that no discussion can ever resolve.

All I can say is that IF we start betting based on our different probability theories, then I'll win (on average). This is because the probabilities I calculate are based on relative frequencies and may be the basis for betting. Whereas, the probabilities you calculate do not relate to the relative frequency with which events take place and, therefore, cannot be the basis of betting without losing money (on average).

I don’t think my way of calculating is different from how all halfers ( there is another position called the double halfer position which I’m not going with)calculate. I also don’t think our disagreement here will apply anywhere else. It’s possible there is a misunderstanding about what I’m trying to say.

About the betting, I think it matters how you apply a bet to the situation. I think she should be told she will only be placing a bet once whether the coin landed heads or tails. Knowledge about how many times she will be asked into a bet shouldn’t effect her belief about how the coin landed.

 

[Dale]

So I just finished the Monte Carlo simulation. I simulated a total of 10,000 flips, of which 4907 came out heads and 5093 came out tails, leading to $\hat P(head) = 0.491 \approx 1/2$. When she was awake there were 4907 heads and 10186 tails so $\hat P(head|wake) = 0.325 \approx 1/3$. She was offered the second bet on Mondays and won the second bet 5093 times and lost it 4907 times so $\hat P(win2) = 0.509 \approx 1/2$.

So I was incorrect about the second bet, which would be the credence of the credence. As you suggested that is indeed 1/2. I was wrong about this, but as I said this is unambiguously not the credence.

The credence is P(head|wake) and is indeed 1/3 as I said and the coin is indeed fair since P(head) is indeed 1/2 as I also said.

 

[hutchphd]

So I was incorrect about the second bet, which would be the credence of the credence. As you suggested that is indeed 1/2. I was wrong about this, but as I said this is unambiguously not the credence.

Just an opinion:
That is the first interesting thing said this entire discussion (More so because @Dale actually had it wrong to begin with!)
The rest of it seems both obvious and silly. Perhaps I am too literal (or stupid) to see the fine points but the fat lady (with apologies to @PeroK) sang in post #4, and I cannot figure out what the ensuing 100+ posts are about. Chacun a son gout I guess but if someone could explain why this is interesting I would appreciate it.

 

[Office_Shredder]

Sorry, can someone remind me what the second bet is, and what she is supposed to be doing?

.
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.

This is my point exactly. Probability is intended to answer the question of "if we do this a lot of times, how frequently does everything happen". If you're not answering that question, then you're not solving a probability.

Like, is I said the answer is it's 80% to be heads, because it's either heads or it's tails, and I say it's 80% to be heads, what would you say to demonstrate that I am wrong?