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

[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.
.

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.