The Sleeping Beauty Problem: Any halfers here?

[Stephen Tashi]

No, I'm saying that she has to pay X every time she takes a bet that is wrong. That's just how betting works.
This is the crucial point-- that's all credence means.

Your'e entitled to make-up you own definition of "credence". I'm talking about the definition that has been cited several times in this thread:

https://plato.stanford.edu/entries/imprecise-probabilities/

This boils down to the following analysis:

Your degree of belief in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.

If the experimenter phrases the question as "What is your credence that it will rain tomorrow" and the consequence of SP's answer is that she will receive $1 if the coin landed heads and lose twice what she offered for the bet if it lands tails then SB's answer should be 1/3. This is the best strategy. That answer is not her credence for the event "it will rain tomorrow". To get her credence for rain tomorrow, she is supposed to buy the the bet "you get $1 if it rains tomorrow". Her credence for rain tomorrow is not measured by the price she would pay for the bet "You get $1 if it rains tomorrow and lose twice what you paid for the bet if it doesn't".

 

[bahamagreen]

Assuming the coin is flipped between explaining the experiment and putting her to sleep...

I am asking about the Wiki version of the experiment according to which she knows what she is going to be asked when interviewed. It is only natural for her to wonder why she could not answer the future interview question immediately after understanding the experiment rather than waiting for an interview. It is also rational for her to wonder and check to see if her interview answer would be different than her answer before retiring Sunday evening. I'm asking if thirders would figure she gets two different answers.

 

[Ken G]

Your'e entitled to make-up you own definition of "credence". I'm talking about the definition that has been cited several times in this thread:

https://plato.stanford.edu/entries/imprecise-probabilities/

I am not making up my own, that is precisely the definition I am using. You pay X every time you bet, that's what it says. You get $1 when you win, that's what it says. That's what I'm saying, that's what is right. Credence never has anything to do with how many times you bet, that's why that element is never mentioned in its definition. In fact, SB is free to bet any number of times she likes, she can bet 10 times in one waking, and 5 times in another. It never affects the credence at all, because none of that will ever affect the fair odds!

 

[Ken G]

It is only natural for her to wonder why she could not answer the future interview question immediately after understanding the experiment rather than waiting for an interview.

And indeed she could, she merely needs to include the information she will have available then, compared to now. (She knows she will be wakened, that's information as proven in the example where there is never any waking for a heads.) The situation is like this. You are playing bridge, and you know there's a 50-50 chance the queen was dealt to either of your opponents. So that's your credence at the start of the hand. You know you won't have to decide which way to do the finesse until later in the hand, so you play the other suits and gather information. Let's imagine you know you will be able to ascertain which opponent has the queen based on how you play the hand (it doesn't matter how you'd actually know that, it's a hypothetical example). It is natural for you to ask what your credence is going to be after you have done that-- the point is, at the beginning of the hand, you have 50-50 credence, and at that later point in the hand, you know your credence will be certain, even as you are thinking at the start of the hand. So it makes no difference when you do the thinking, what matters is what information you know you are going to have available at the time you ascertain your credence. This is a perfectly practical consideration, it is the reason airline pilots are trained to make whatever decision will allow them to gather more information before they have to make the life-or-death decision. It's the most practical consideration there is-- credence at a given moment is based on available information at that moment.

I'm asking if thirders would figure she gets two different answers.

And the answer is yes, just like the bridge player, just like the airline pilot. To get the right credence, you don't need to have the information the whole time, you only need to know, the whole time, that you will have information when you ascertain the credence.

 

[Marana]

Use four volunteers, and the four cards I described before (with (H,Mon), (H,Tue), (T,Mon), and (T,Tue) written on them). Deal the cards to the four, and put them in separate rooms. Using one coin flip, and waken three of them on Monday, and Tuesday. Leave the one whose dealt card matches both the day, and the coin flip, asleep. Ask each for her confidence that the coin matches her card.

Obviously, if you show each Beauty her card, her answer has to be the same as the original Beauty's. Since it is the same regardless of what card is dealt, you don't have to show it to any of them. If you don't show it to any of them, you can put all three awake Beauties in a room together to discuss their answers. All have the same information, so all answers have to be the same. Since exactly one of the three has a card that matches the coin flip, that answer must be 1/3.

You seem to be making a big assumption without explicitly stating it.

Suppose there are 1001 beauties: 1 winner who wakes up 1000 days in a row, and 1000 losers who wake up once in that time. Each thinks on sunday "There is a 1000/1001 chance that I wake up next to the winner."

Your assumption is that not only must they answer the same way to "am I the winner?", but that the correct way to compute probability is to divide it up equally among those with symmetric information. That is, even if I correctly believe there is a 1000/1001 chance that I will wake up next to the winner, when I actually do wake up next to someone I should split the probability evenly, giving myself a 1/2 chance of being the winner.

But you haven't justified that assumption. I argue that, if I am correct in believing I have a 1000/1001 chance of waking up next to the winner, then when I wake up next to someone it can make me think they are the winner. I am no longer indifferent to them because they are across from me when I wake up, which I didn't know would happen. But I did know I would wake up on the same day as myself.

The fact that they have symmetric information that leads them to believe I am the winner is odd, but I don't see why it requires me to "divide the probability evenly."

The bet is implied by the definition of credence. So it is every time she is asked about her credence, which is every interview.

They were stated in the original problem. Beauty is asked about her credence in each interview so the implied bet is necessarily offered each interview also.

This is just false. The definition of credence does not provide the ability to see into the future or the forgotten past.

You are adding that ability because credence is extremely hard to define in the actual problem. But sleeping beauty considering a bet being offered now does not imply that she can rely on the existence of specific past or future bets. That simply isn't in the definition, and it has nothing to do with cheating on the part of the experimenters.

It may be that we can't come up with a coherent way to apply the definition of credence to this situation. Or maybe we can say that she accepts bets on sunday when she has P(H) = 1/2. Or maybe we can use reflection to wednesday at noon when she has P(H) = 1/2. Or maybe we should consider "surprise", in which case the lottery example is compelling to me. The thirder answer means that sleeping beauty can become arbitrarily confident that she won an arbitrarily unlikely lottery.

A is Beauty is awakened during the experiment (i.e. with amnesia, being interviewed, and being asked her credence that it is heads).

Are you under the impression that beauty always has amnesia when asked her credence? The experiment goes like this:

1/2: sunday coin flip heads -> monday interview
1/2: sunday coin flip tails -> monday interview -> (amnesia regarding monday) -> tuesday interview

Note that there has been no amnesia before the monday interview, and that there is never any amnesia that severs the causal link between the sunday coin flip and the current interview.

This is a crucial point. It is not enough for sleeping beauty to be told the rules when she wakes up, even if you tell her it is the first time the experiment has ever been performed. If she has total amnesia, then it is too late. Her awakening is already selected by the time you explain, and it becomes a different problem. The unbroken causal link from sunday is essential.

 

[PeterDonis]

I am not making up my own, that is precisely the definition I am using.

The definition you are using is in ordinary language, and as I've already pointed out several times in this thread (though a while ago now), ordinary language is vague. That is what you and @Stephen Tashi are illustrating. As soon as you both agree on the actual mathematical problem being posed, you agree on the answer. So the only dispute left is about vague ordinary language. Such disputes are pointless IMO.

 

[PeterDonis]

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