Which is why I had included the proviso that the (rare) opportunities be proportional to the number of days the hostage is held captive. Under those conditions, they carry no information to the hostage. — Pierre-Normand
It's quite straightforward that P(Dice roll 6|opportunity to escape) > P(Dice roll 1-5|opportunity to escape) — Michael
Indeed, which is basically the 'thirder' solution (in this case, the 5/11er solution). — Pierre-Normand
Likewise, enabling Sleeping Beauty to bet on H on each awakening provides no information to her, provided only the payouts are delivered after the experiment is over. — Pierre-Normand
I don't understand the connection between Sleeping Beauty's credence that the coin landed heads and the tracked frequency of heads-awakenings. It's a non sequitur to claim that because tails-awakenings are twice as frequent over repeated experiments then a coin toss having landed tails is twice as likely in any given experiment.
Sleeping Beauty is being asked "in this current, one-off experiment, what is the probability that the coin I tossed on Sunday evening landed heads?".
She's not being asked to guess if it's heads or tails and then being rewarded for each successful guess.
Her choice of guess in the latter has nothing to do with what her answer would be to the former.
If I were Sleeping Beauty I would answer "1/2" and guess tails.
Can you give actual numbers? — Michael
Suppose there is a 0.01% chance to find an opportunity to escape on any given day held captive regardless of that day being the only one or one among six in a kidnapping event. Finding such opportunities doesn't yield any updating of credence. — Pierre-Normand
It does. — Michael
What if there is a 1% chance that the tulip is red on any given awakening day? Would that make any difference? — Pierre-Normand
Yes. The probability of it being red if heads is 1%. The probability of it being red if tails and Monday is 1%. The probability of it being red if tails and Tuesday is 1%. The probability of it being red if tails is 1 - 0.99^2 = 0.199.
The probability of it being red if tails is greater than the probability of it being red if heads, therefore if it's red then it is more likely tails. — Michael
Consider this analogy: you're entering a movie theater where there's an even chance of a double feature being shown. There's a one percent chance that any given film will feature Rex Harrison. Suppose you see Harrison featured in the first film. Does that increase your credence that there will be a subsequent feature? — Pierre-Normand
If you think about it, Lewis's notion—that Sleeping Beauty can conclude from knowing it's Monday that a future coin toss is more likely to yield heads with a 2/3 probability—is already rather puzzling. — Pierre-Normand
No, but if I walk in not knowing if it’s the first or second then my credence favours it being part of a double feature. — Michael
Sorry, misunderstand the movie example. It’s a different answer if I only get to walk into one film, which would be comparable to Sleeping Beauty only waking on Monday (or Tuesday) if tails. — Michael
The main point is that seeing Rex Harrison being featured (while knowing that 1% of the movies randomly being shown in this theater feature him) doesn't impact your credence in this movie being part of a double feature. — Pierre-Normand
That's only because I walk into one film. If I'm given amnesia and walk into the second film (if there is a second film) then it affects my credence.
It's exactly like my scenario with the coin toss and prizes. If heads then the car is the possible prize, otherwise the motorbike is the possible prize. If a car then a single coin toss determines if I get it (if heads), if a motorbike then two coin tosses determine if I get it (one head is enough to win). — Michael
In your scenario, the nature of the prize is conditioned on the coin toss results. — Pierre-Normand
Then forget the nature of prize. If I know that I’ve won a prize my credence is that the first coin toss landed tails. — Michael
However, in the scenarios with Sleeping Beauty and the prisoner, merely being presented with an opportunity to bet or escape does not give them any new information about the outcome of the coin toss (or throw of the die). They must decide how to take advantage of this opportunity (by choosing to carry the torch or the plank, or choosing what safehouse address to communicate to the police) before gaining any knowledge about the success of the attempt. The offering of the opportunities carry no information and provide no ground for updating credences. — Pierre-Normand
She can and should use known priors to condition her credence, and one such prior is that she is more likely to win a prize/have the opportunity to escape if tails/a dice roll of 6. As such, if she wins a prize or has the opportunity to escape she should condition on this and her credence should favour tails/a dice roll of 6, otherwise her she should condition on not winning a prize or having the opportunity to escape and her credence should favour heads//a dice roll of 1-5. — Michael
And if she's guaranteed the opportunity to escape each day her credence should favour a dice roll of 1-5.
But honestly, all this talk of successes is irrelevant anyway. As I said before, these are two different things:
1. Sleeping Beauty's credence that the coin tossed for the current, one-off, experiment landed heads
2. Sleeping Beauty's most profitable strategy for guessing if being asked to guess on heads or tails over multiple games
It's simply a non sequitur to argue that if "tails" is the answer to the second then "1/3" is the answer to the first.
So, she should be carrying a plank and end up being eaten by lions on 6 out of 11 escape attempts? — Pierre-Normand
The manner in which (1) is stated suggest that Sleeping Beauty is referring to the wide centered possible world spanning the whole experiment run. In that case, her credence in H should be 1/2.
The second one makes it rational for her to rely on her credence regarding narrow centered possible worlds spanning single awakening episodes. There indeed isn't any entailment from the suitability of one framing of the question from (1) to (2) or vice versa. The two sentences concern themselves with different questions. — Pierre-Normand
5 out of every 6 victims escape. I count by participants, not by escape attempts. I think it's more reasonable. — Michael
Since on my approach probabilities track frequencies, even if there is just one kidnapping event, the hostage's chances of survival are 5 in 11 whenever an escape attempt occurs. — Pierre-Normand
And that first question is the premise of the problem. Sleeping Beauty is asked her credence that the coin landed heads. That's it. She's not being asked to consider the most profitable betting strategy for multiple games. — Michael
So then there are two different ways to reason with nothing to prove that one or the other is the "right" way? — Michael
I think you would benefit from reading Groisman. — Pierre-Normand
There are two ways to reason:
1. of all interviews are 100 heads in a row interviews, therefore this is most likely a 100 heads in a row interview
2. of all participants are 100 heads in a row participants, therefore I am most likely not a 100 heads in a row participant
I would say that both are true...
That's correct since events that happen in the world don't come flagged with sign posts that say: "the current event begins here" and "the current event terminates here." How credences in the probabilities of events are assessed depend on the way those events are individuated and this can be dictated by pragmatic considerations. — Pierre-Normand
But as this debate has gone on long enough and I don't think I have the energy to continue it much more, I'm happy to just say that both 1/2 and 1/3 are correct answers to distinct but equally valid interpretations of the question. — Michael
There are two Sleeping Beauties; one will be woken on Monday and one on both Monday and Tuesday, determined by a coin toss.
What is their credence that they have been or will be woken twice?
Rather than getting back into the nitty-gritty, I'm thinking about the stuff I posted a while back, the possible self slices and all that. — Srap Tasmaner
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