Sleeping Beauty Problem
For those who haven't seen it, here is the wording:
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details. On Sunday she is put to sleep. A fair coin is then tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a dose of an amnesia-inducing drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday. The drug makes sure that she cannot remember any previous awakenings during the course of the experiment, but she will retain the ability to memories gained after the experiment is over.
Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"
There doesn't appear to be a mathematical consensus on the correct answer. Obviously, from the experimenter's perspective, the odds are exactly 50/50 for heads/tails. From Sleeping Beauty's perspective, however, she is awoken twice as many times, on average, for a tails toss than a heads toss. For example, from Wikipedia, if you toss the coin 1000 times, she will be awoken 500 times on Monday for a heads toss, 500 times on Monday for a tails toss, and 500 times on Tuesday for a tails toss, giving a total of 2/3 of the time she is awoken, tails was thrown.
However, as an outside observer, we know it had to be 50/50 in the actual toss, so where does the difference enter? I think I'm in the 1/3 camp, but can't fully justify it. What if she is awoken once after a heads toss, and 100,000 times after a tails toss, should she believe with near certainty that tails was thrown? Seems almost like she should be able to correctly guess 2/3 of the time that it is Monday, but the number of times she is awoken should have no effect on the perceived likelihood of what was tossed.
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 observ
Not at all clear, nor is it clear how the million-sided die could have any impact on this situation. What if I get a deck of cards out and do a magic trick, or bullseye a womprat in my T-16? She's always going to go through the process of waking up and not knowing what day it is. Why does the fact she's woken up update anything? Let's update our calendars, but it's still a 50% chance she's woken up on a monday for the one and only time, a 25% chance she's woken up on a monday is going back to sleep, and a 25% chance she's woken up on a tuesday and is having a good laugh at us behind our backs. She experiences it the same way we do. No part of her experience should - as far as I understand it - have any impact on the underlying probabilities. Barring some heretofore unproposed causal mechanism, I'm struggling to understand how her waking up could have an effect on something that's already happened.
If the above betrays a lack of understanding of Bayes, on my part, please lmk and I'll look into what I'm not getting.
[quote=wazz]
nor is it clear how the million-sided die could have any impact on this situation.
[/quote]
https://en.wikipedia.org/wiki/Bayes%27_t...
Here's an extreme example. A coin is tossed. If heads, John rolls a thousand-sided die with 999 sides colored blue and 1 side colored red. His die color is reported to you. If tails, Jill rolls a 1000-sided die with 1 side blue and 999 sides red. Her color is reported to you. Only the die color is reported, not who rolled it.
Your credence for the coin flip before a die color is reported to you is 50-50. How is it possible your credence for the coin flip can change due the report of a die color from a die roll that was done after the outcome of the coin flip had already been determined?
When Baye's Theorem first came out there were people who argued against it based on the thinking that a later event can't change the outcome of a previous event, so can't change the likelihood of a previous event. Their thinking was that Baye's Theorem reverses the order of cause and effect so must be wrong.
PairTheBoard
there are so many smart people on twoplustwo. Quora is what I use. there are at least 3 million geniuses who use quora.
most people don't even know a lot of expert poker players are serious mathematicians, mind readers or whizzes at computer science etc.
I tried to read the puzzle, but I couldn't figure out what it even meant, partly because I'm autistic with problems reading.
Flunkie who joined Ike Haxton/Scott Seiver Brown Chapter of the Aepi. fraternity which played a role in ruining my life.
there aren't even 3 geniuses on quora, let alone 3 million
most answers are superficial, those which aren't are often full of inaccuracies
there are so many smart people on twoplustwo. Quora is what I use. there are at least 3 million geniuses who use quora.
most people don't even know a lot of expert poker players are serious mathematicians, mind readers or whizzes at computer science etc.
I tried to read the puzzle, but I couldn't fig
I wouldn't trust quora. I'd trust the output of an LLM like copilot over quora. i doubt there's more than 100,000 'geniuses' there. Like 2p2, a lot of people who are in love with their own ideas, and don't pay enough attention to humility or cognitive biases. A lot of cultural bias in there too. You have to be very careful with an LLM but they can be useful research tools.
To break the puzzle down to simpler terms, due to the process of your memory being partially wiped, you don't know if you're being woken up on monday or tuesday; which supposedly tells you information about a coin being tossed. if it doesn't make sense to you that an event in the future can somehow change the probability of an event in the past, that's not just you.
I don't "trust" Quora always.
The beauty and simplicity of the website is you learn who to listen to, and why. It's a community, a bit like this website.
if it doesn't make sense to you that an event in the future can somehow change the probability of an event in the past, that's not just you.
If you have incomplete information about whether a past event happened, does it make sense to you that finding out a certain current event happened could provide you with additional information about whether the past event happened?
PairTheBoard
I don't "trust" Quora always.
The beauty and simplicity of the website is you learn who to listen to, and why. It's a community, a bit like this website.
one of the most prolific writers on the site,
literally has answers with thousands of upvotes declaring such nonsense as all lobsters originate in the caribbean as eggs and ride the currents north to grow and live, or that babe ruth has the record for caught stealing, or that toronto attracts more tourists than anywhere else in the world
one of the most prolific writers on the site,
literally has answers with thousands of upvotes declaring such nonsense as all lobsters originate in the caribbean as eggs and ride the currents north to grow and live, or that babe ruth has the record for cauWhich is presumably why YeBlessChildren has taken the approach of learning who to listen to and why
If you have incomplete information about whether a past event happened, does it make sense to you that finding out a certain current event happened could provide you with additional information about whether the past event happened?
PairTheBoard
Yes, but it's far from clear to me that that's what's happening here.
If you have incomplete information about whether a past event happened, does it make sense to you that finding out a certain current event happened could provide you with additional information about whether the past event happened?
The thing is, I agree with you when it comes to the common 1/3er argument that SB gains information by merely waking up. The information supposedly being that she's in an awakening. Jason1990 also disagrees with that argument as he states in his summary at the end of post 662 - which I show again below. So, support for the 1/2er view.
I'm also not entirely comfortable with the rest of Jason1990's analysis which supports the 1/3er position. It may be the prior probability of SB's total waking experience is actually zero. Like picking a number between 0 and 1 from a uniform distribution. Any number you actually pick had prior probability zero of being picked.
I'm also not entirely comfortable with Jason1990's 20-sided die scenario. For any outcome of the die roll, SB draws the same conclusion. One that supports the 1/3ers. So why bother rolling the die? She knows ahead of time what conclusion she will draw from its outcome. Jason1990 points out that this logic is not supported via a total probability application in the probability model. Of course he's correct about that. But I'm afraid I just can't shake the power of my common sense that does support it.
All of which leaves me back with SB exiting the experiment to the freezer under heads and to the dessert under tails. When SB is in an awaking, regardless of how she feels about the matter now, the one thing she knows is that when the experiment is over there's a 50-50 chance she's going into the freezer.
==================
From 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
