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neq1 comments on Conditioning on Observers - Less Wrong
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Another Rival Intution Pump
Suppose that for exactly one day every week you sleep during the day and wake up in the evening, and for every other day you sleep at night and wake up in the morning.
Suppose that for a minute after waking, you can reason logically but cannot remember what day it is (and have no way of telling the time).
Then during that minute, surely your subjective probability of it being morning is 6/7.
OK now let's change things up a little:
At the beginning of every week, a coin is flipped. If heads then rather than having 6 days diurnal and 1 day nocturnal, you just have 1 day nocturnal and six days in hibernation. If tails then you have 6 days diurnal and 1 day in hibernation.
Then surely the addition of the coin flip and the hibernation doesn't change the fact that (for any given awakening) you have a 6/7 probability of waking in the morning.
I think the coin flip does change things. In fact, I don't see why it wouldn't.
In case 1, you know you are somewhere along a path where you will wake up on one night and wake up on 6 mornings. You can't determine where along that path you are, so you guess morning has probability 6/7
In case 2, there is a 50% chance you are somewhere on a path where you wake up once at night (and never in the morning), and a 50% chance you are somewhere on a path where you wake up on 6 mornings and 0 nights. So, probability it is morning is 1/2.
So even though, in the long run, 6/7 of your awakenings are in the morning, and you have (for that first minute) no information to help you work out which awakening this is, you still think that on any given awakening you ought to feel that it's just as likely to be morning as evening?
Sure you can bite the bullet if you like, but quite frankly your intuitions are failing you if you can't see why that sounds strange.
Not all awakenings are equally likely. 50% chance it's one of the 6 morning awakening. 50% chance it's the one night awakening.
Suppose two weeks' worth of coins are tossed ahead of time.
Then with probability 1/4, you will wake up twice in the evening. With probability 1/2, you will wake up 6 times in the morning and once in the evening. And with probability 1/4 you will wake up 12 times in the morning.
Then by your logic, you ought to say that your probability of waking in the morning is (1/4)x(0/2) + (1/2)x(6/7) + (1/4)x(12/12) = 3/7 + 1/4 = 19/28, rather than 1/2 if the coins are tossed 'just in time'.
How can whether the coins are tossed in advance or not change the subjective probability?
By neq1's previous reasoning, there's 50% chance of waking in the mornings and 50% chance of waking in the evening for any particular week. That is the case whether the coins are tossed in advance or not. The probability of a particular morning awakening would be 1/12.
I'm not sure where you got your (6/7) figure for neq1's calculations.
neq1 admits that in my original scenario, before I introduced the coin and hibernations, you have a 6/7 probability of waking in the morning. The case where one of the two coins is heads and the other is tails is equivalent to this.
Sorry, I'm not following. What are you doing with these two weeks' worth of coins?
In the situation I described previously, at the beginning of each week a coin is tossed.
What I'm doing is saying: Suppose week 1 AND week 2's coin tosses both take place prior to the beginning of week 1.
Your original question was about one week, not two (I thought).
Are we just doing this twice? What happens between the weeks? Do they know the experiment has started over?
Could be, or could be a great number of weeks. Shouldn't make any difference.
Nothing (except that, if necessary, the next week's coin is tossed.)
They know it will start over, and once the 'minute of confusion' has passed, they become aware of all that has happened up to now. But during the 'minute of confusion' they only know that 'an experiment is in progress and it is week n for some n' but don't know which n.
Once you go more than 1 week it's not the sleeping beauty problem anymore. Half the time she's woken up once at night, 1/4 of the time she's woken up 6 times in morn and once at night, 1/4 of the time she's woken up 12 times in morn. This doesn't have to do with when the coins are tossed. It's just that, if you do it for 1 week you have the sleeping beauty problem; if you do it multiple weeks you don't