Sleeping Beauty Problem

The correct answer is 1/3 because the question is asking about EV, not about probability

Tails grants twice the EV (2x the number of wakeups) as heads.

Run the experiment twice, 1 heads, 1 tails.

Sleeping beauty is asked each wakeup what she thinks the coin is:

if she says heads, she's wrong twice and right once

if she says tails, she's right twice and wrong once

The question is about expected value, not probability

In other words, think of this way. Instead of 2 wakeups on tails, imagine 2 trillion wakeups. So 1 wakeup for heads and 2 trillion wakeups for tails.

Now run the experiment and ask sleeping beauty if she'd like to buy heads for 50c? No? You don't want to buy heads for 50c? So then you think the probability must be lower

When the coin comes up heads, sleeping beauty is only asked the question 1 time. When it comes up tails, she's asked it two trillion times.

The probability of the coin didn't change, she is just asked the question more often when it's tails.

That's interesting! Actually, the question is about neither "probability" nor "expected value". Quoting the OP, the question is about "credence".

"Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"

According to Google, "credence" means ...
"belief in or acceptance of something as true"

So you would argue that in this context "credence" means "expected value"?

PairTheBoard

Yes, it does, because she is asked the question more often when the coin is tails than when it is heads.

Run the experiment twice at neutral probabilities (1 heads/1 tails)----> get 1 heads wakeup and 2 tails wakeup

So she's asked the question 3 times.

If she says her credence is 1/3rd, she'd be correct every time

In other words, if sleeping beauty were asked the question: would you buy heads for 50c? she shouldn't accept this bet. Because heads is only worth 33c.

Maybe her "credence" should not be based on how she should bet on a gamble you've proposed which was not included in the original problem. Maybe her "credence" should be based on what she predicts will be seen on Wednesday was the outcome of the flip.

Repetitions of a prediction have no bearing on whether it proves accurate.

PairTheBoard

Credence is based on how you should bet. All else is BS.

It's structured related to the EV, aka the outcome. The outcome is 1/3rd heads 2/3rds tails because she gets asked the question twice when it's tails and only once when it's heads

Like I said, just make the question: sleeping beauty, would you buy heads for 1:1 odds? No? Then it's not 1/2

The "outcome" is the objective fact for whether the coin landed heads or tails. The structure of the "gamble" is having her het once if the outcome is heads and twice if the outcome is tails - with no memory of the first bet when she makes the second bet under the outcome of tails.

The "gamble" is rigged. She knows it's rigged, and she knows how it's rigged. That she has this rigged gamble imposed on her while she's inside the box of sleeps doesn't change her prediction for what outcome of the flip will be revealed when she comes out of the box. That prediction is her true "credence" not the EV of the rigged gamble.

PairTheBoard

Maybe it is, but what I have been meaning to ask is this. Is there a recognized interpretation of probability where the probability of an event is defined as the total money bet on the event divided by the total money bet on the experiment, and any other definition of what probability means is illusory?

The frequentist interpretation of the probability of an Event is the long run frequency the Event happens over numerous repeated trials of the experiment.

From Wiki -
"Bayesian probability (/ˈbeɪziən/ BAY-zee-ən or /ˈbeɪʒən/ BAY-zhən)[1] is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation[2] representing a state of knowledge[3] or as quantification of a personal belief.[4]

The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses;[5][6] that is, with propositions whose truth or falsity is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probabiity.
-------------

I'm not sure what the name might be for what you're describing. Maybe the "Mutual Betting Pool" interpretation of probability. For example, the horse racing odds at a hypothetical race track where there's no money removed from the mutual pool for expenses or fees or whatever.

Regardless, remember the original problem doesn't ask for a probability.
The OP -
Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"
---------

PairTheBoard

Suppose this "experiment" is repeated every week. Every week SB makes the prediction the coin lands heads. She expects her prediction to be proved accurate 50% "of the time". She repeats that prediction inside the experiment as needed. Repetitions of her prediction have no bearing on its accuracy. Over numerous trials of the "experiment" her prediction proves accurate in about 50% of the experiments, i.e. about 50% of the time.

I think this should be her "credence", this is her "credence", and all else is BS.

PairTheBoard

If you treat SB's "credence" as a prediction then forcing her to repeat it when it's already been determined as being wrong doesn't change the prediction's accuracy any more than a prognosticator who repeats his prediction numerous times when it's already been determined as coming true.

What the 1/3'ers are doing is treating her "credence" as a willingness to take all bets on it, even ones that are "past-posted".

Take your pick.

PairTheBoard

Too much to do anything but grunch. It will always be 50/50, because you're tossing a coin and you're doing it once. From her perspective, there is a 50% chance it's just the once, and a 25% chance she's being woken up on the memory-wiping day, and a 25% chance she's being woken up on the final day. This is a classic case of confusing possibility with probability. It feels like 1/3 vs 2/3 because there are 3 options, but they are not weighted equally.

EDIT: the bit that drives this home is in the Veritasium video where he asks what happens when you do the same experiment with brazil vs canada 80/20 but wake them up on a canada win twice in a row. If you then gave them a betting stub they could cash in once on a canada win, they would only win that once. The odds of a Canada win haven't changed. Your experiencing of it has, but the odds don't change through your experience, just how they're divided up amongst your possible selves.

The problem with this view is that if she wakes on Monday and is told that it's Monday, she can't help but think it's twice as likely she got there via a heads than a tails. One way to sidestep this problem is to just "not know" more than "it's either Monday or Tuesday" if she awakes and is told the coin was tails.

The best technical analysis in this thread is made by jason1990, a professional probabilist. He tackles the problem with applying probability theory to problems with "indexicals" as in Sleeping Beauty. You can jump into it with posts 661 and 662 with call back to post 281.

PairTheBoard

I'll check that out but it's not twice as likely, it's equally likely, because we're dealing with probability, not possibility.

Still not had chance to read those posts but here's how the logic goes:

the {actual chance of X} remains the same, but the {chance of X given she's waking up} is proposed to be different because her having woken up applies equally. But the problem is that her waking up is causally connected to X in the wrong direction. It is X that causally comes before {waking up}. Her waking up can have no backwards influence on that event. It seems trivially easy to say that her odds remain 50% waking up {first and only time}, 25% {first of 2} and 25% {second of 2}. I'll look through the thread to see if I can see anything that could convince me otherwise.

This is driven home by the football match argument. I feel like Bayes would hold his hands up and say 'nothing to do with me' and point us in the direction of the boy who cried 'wolf', and the wolf is barking up the wrong tree, and this is the actual time it's the wolf, but he's still barking up the wrong tree.

Blimey. That was a lot of stuff that was dense in detail and way over my head. It feels like he agrees with me.
The problem feels like it stems from 'she can't help but think it's twice as likely she got there via a heads than a tails'. Only if we allow for her to think fuzzily.

Sorry to not bunch my posts up together and into one. The Monty Hall problem has the wolf barking up the right tree. In that there is a causal mechanism in the form of elimination of possibility that is not present here. In order to apply a filter to past events like this, there needs to be a causal mechanism. She wakes up and says 'the odds are 50/50 that a heads was thrown, and that therefore this is a monday and I'm not going back to sleep. There's a 25% chance this is a monday and I AM going back to sleep. And a 25% chance this is a tuesday and I'm not going back to sleep. My having woken up can not have had any influence on the odds of a cointoss, and the third indistinguishable possibility splits up one of the existing possibilities and no information is transferred to me by virtue of having woken up - which I was always going to do.'

Done, lock the thread

Just to clarify, because this is slightly confusing.
Yes, if she's told it's monday, then it's twice as likely she got there via a heads than a tails. But if she's not told it's monday, we have to add in the 1/4 chance it's a tuesday, giving us the entire probability space. It's twice as likely i.e. thirdsed when we chop off a quarter of the whole. I.E. it's not a problem that has to be sidestepped, it's a problem solved by not creating it in the first place.

I'm afraid that is a problem because it's clearly just wrong. An easy way to see this makes no sense is to wake SB on Monday as usual but flip the coin Monday night instead of Sunday night. That doesn't change anything because the only effect caused by the coin's outcome is to decide whether or not there will be an awakening on Tuesday.

When flipping the coin Monday night your statement above refers to the outcome of a coin flip that hasn't happened yet.

I urge you to read some of the thread. Especially the posts I suggested above.

PairTheBoard

This is a very confusing proviso you've added. I'm now tempted to change my answer back again to 50%. Because if it came up tails, then this is just the first of two differentiable waking experiences, and now it's just the case that she's flipped a coin and half of the time she's going back to sleep and will forget this and half of the time she won't.

It seems obvious that if the true odds of a thing are 50/50, then there must be something powerful going on that could change those odds. What is the mechanism behind which the act of waking up - something that happens in this experiment regardless - could go back in time and change the odds of something? What if I bought 100 lottery tickets and told someone 'if I win, keep putting me back to sleep and wiping my memory and waking me back up again the next day' - does that improve the odds of my winning? Or does it just slice up what I assume functions like an indexical - my experiencing of that reality? Reality heads is smooth, reality tails is sliced into 2, but it's not all coming from the same place - reality tails being sliced into 2 does not draw on reality heads, which is unaffected. Cannot be affected by the waking or not of an event that happens afterwards.

If you can find a slightly more simple explanation of why the above is wrong and why it's clearly a third, I'd be very open to learning. I was wrong about Monty Hall originally and I could easily be wrong about this too

I suggest you study the posts by jason1990 I mentioned above. He explains what "indexicals" are and the difficulties they pose for probability as an extension of formal logic. He then constructs a formal probability space for Sleeping Beauty which does not involve indexicals. He then looks at what happens if SB is allowed to "self locate" herself by way of observing details of where she's at such as rolling a 20 sided die and observing the outcome. This gives her a way to distinguish "this awaking" from the others. Under this non-indexical formal probability space such self-locating motivates a 1/3 credence.

However this is his summary at the end of his post 662.

[quote=jason1990]
From a practical perspective, Sleeping Beauty doesn't even need a die. Perhaps she wakes up and observes that the interviewer is standing exactly 32.1 inches from her when he asks her the question. Or perhaps she observes that she has an itch on her right cheek exactly 1 minute and 18 seconds after waking. If her prior credence for these events happening on both days is much smaller than her prior credence for them happening on a given day, then her posterior credence for heads will be close to 1/3.

But of course, all of these modifications seem to violate the spirit of the problem, which is that Sleeping Beauty experiences absolutely nothing that could even possibly distinguish one awakening from another. In that case, if credence is understood in the context of formal probability theory, and if her credence for heads was 1/2 before the experiment, then it would have to be 1/2 during, since she obtains no additional, objective information.
[/quote]

PairTheBoard

I'm afraid that doesn't shed light on it for me. It's going to be too much work for me to study and learn that formal logic about indexicals and I don't care that much. I still don't understand how he's not then concluding in bold that it is still 1/2, no matter what die she rolls or how that could have any influence on.

Is this not a thing we could somehow test out with AI? I assume this thread has been updated in the era of chatgpt and asked what it thinks

I think chatgpt would write a nice high school paper about the current thinking on the topic. Basically something like a Wiki entry. Nothing that would indicate any original analysis.

Maybe this will give you some feeling for what's going on with jason1990's analysis:

Suppose you're an outside observer. You know how the experiment works and you know SB rolls a million-sided die whenever she awakens. You don't know the outcome of the coin flip. On Wednesday you ask SB, "did you roll the number 417,953 during the experiment?" Suppose SB says "yes". Aren't you going to do a Bayesian update to your prior probability for heads from 1/2 to 1/3? Isn't it clear that such an unlikely event was about twice as likely to have happened under tails as heads? You might want to read up on Baye's Theorem.

PairTheBoard