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

[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.

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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?

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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?

 

[Moes]

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.

Can you give me your opinion on this?

 

[Moes]

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.

let me know what you think about my last message.

 

[Moes]

I would like to take an opportunity to thank everyone in this discussion. This thread is really wonderful! All of you are amazing! I thank everyone for their time.

 

[Dale]

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

Look, I am quite open minded here, I have been willing to write computer code for a Monte Carlo simulation explicitly to find my own mistakes. That is certainly more willingness to understand your view than you have shown towards understanding mine. So please stop with this sort of comment. You say things like “your going wrong” while telling me “Try to understand my view” and implying that I am only looking to argue. This basically announces your closed-mindednesses while demanding that I agree with you to demonstrate my open-mindedness. I cannot be accused of closed-mindedness when I have gone out of my way to write code that in the end showed a mistake I made which I accepted. Please keep your own advice and don’t give it to me.

I disagree with your argument. That does not mean I misunderstand it in any way.

I guess the real issue is more about the concept of credence itself.

I think there is a problem with the way your defining credence.

Ok, so we do now agree on the central issue. It is regarding the definition of credence.

I posted two references about the scientific definition of credence. So it is not how I am defining credence, it is how the community defines it. If you disagree on the definition then please post your source providing the scientific definition of credence. (A dictionary is not such a source as scientific terms often have different meanings than the same word used non-scientifically e.g. field = open area with grass vs field = a function on spacetime)

This is why I think my case of the second bet is a more accurate way to calculate her credence.

Regardless of the definition of credence, your second bet concept is unambiguously not relevant. Your second bet is explicitly only offered on Mondays. But the credence she is asked to state in the scenario must be obtained on both Mondays and Tuesdays. On Tuesday she will not remember her credence of Monday’s credence, so it cannot be said to represent her credence on Tuesday.

No, the credence of Monday’s credence is not the credence. Particularly given that it is deliberately designed to miss a valid part of the experiment. That argument is a non starter, and I am not open to further discussion of it.

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

Science often changes definitions of words. That is why you need to find a source dealing specifically with the scientific term if you don’t like my sources. I am open to that, but not to general dictionaries.

 

[Moes]

Look, I am quite open minded here, I have been willing to write computer code for a Monte Carlo simulation explicitly to find my own mistakes. That is certainly more willingness to understand your view than you have shown towards understanding mine. So please stop with this sort of comment. You say things like “your going wrong” while telling me “Try to understand my view” and implying that I am only looking to argue. This basically announces your closed-mindednesses while demanding that I agree with you to demonstrate my open-mindedness. I cannot be accused of closed-mindedness when I have gone out of my way to write code that in the end showed a mistake I made which I accepted. Please keep your own advice and don’t give it to me.

Sorry if this is the way I sounded I was just trying to make sure we were both staying open minded. I believe I am . I would also just point out that I wrote that message before you came out with the results of the simulation.

Regardless of the definition of credence, your second bet concept is unambiguously not relevant. Your second bet is explicitly only offered on Mondays. But the credence she is asked to state in the scenario must be obtained on both Mondays and Tuesdays. On Tuesday she will not remember her credence of Monday’s credence, so it cannot be said to represent her credence on Tuesday.

No, the credence of Monday’s credence is not the credence. Particularly given that it is deliberately designed to miss a valid part of the experiment. That argument is a non starter, and I am not open to further discussion of it.
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What I am not understanding here is that you are saying the second bet is not relevant, but you also agree it does give us the credence of the credence. To me this makes it relevant.

I guess this just takes us back to our argument about the definition of credence. Which again, I’m not sure it’s possible to have such a definition. The sources you gave I don’t think took away the word belief from the definition of credence .

I would point out that your dictionary probably doesn’t have a word for the definition that I’m looking for which I find interesting.

As a side point I would point out that I’m pretty sure PeroK disagrees with you about certain things and I would like to know what your argument is about.

The main point is we are coming out that our argument is in language. It would be interesting if this is what the thirder and halfer argument is about.

 

[Dale]

Dale agreed to the following statement:

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

I did not agree to that. The phrase “while Sleeping Beauty is woken up” is a condition. That means that a conditional probability is being calculated.

The notation for conditional probability is $P(event|condition)$. So here the condition is “Sleeping Beauty is woken up” which I denote as $awake$ and the event is “coin landed heads” which I will denote as $heads$. So what you wrote is $P(heads|awake)$, which I showed is $1/3$ and not $1/2$ with the Monte Carlo simulation.

What I am not understanding here is that you are saying the second bet is not relevant, but you also agree it does give us the credence of the credence. To me this makes it relevant.

That is nonsense. If I am asked to calculate the square of 3 then the answer is 9. The fact that the square of the square is 81 is not relevant. And in your case it is even worse since you are removing Tuesday before calculating the second credence. So it is more like being asked for the square of 3, finding 9, then subtracting 1 and squaring that to get 64. It is not relevant. The procedure makes no sense, and the only reason you are suggesting it is because it gives the number you want.

But worse than that, your credence of the credence function only returns two values when three are requested. There is simply no justification to treat a procedure that generates only two outputs as though it were one that generates three. If you were to write a computer program that way it would not even compile.

 

[Dale]

I’m not sure it’s possible to have such a definition.

It certainly is possible to have a valid scientific definition of credence (two of which have already been provided). What seems unlikely is that we will find a scientific definition that gives the result you want. But that is itself an indication that your argument is wrong. The Sleeping Beauty scenario is posed in terms of credence, so if credence were an undefinable term that would imply the scenario itself is fundamentally meaningless.

At this point I am going to close this thread. If you find a source with a scientific definition of credence that you will accept then we can reopen it to work through the problem under that definition. Until then, there is nothing to say other than to point back to the results already shown under the standard scientific definition.