## Thursday, October 16, 2014

### Continuous Sleeping Beauty

A coin is tossed without the result being shown to you. If it's heads, you are put in a sensory deprivation chamber for 61 minutes. If it's tails, you are put in it for 121 minutes. Data from your past sensory deprivation chamber visits shows that after about a minute, you will lose all track of how long you've been in the chamber. So now you find yourself in the chamber, and realize that you've lost track of how long you've been there. What should your credence be that the coin landed heads?

Why is this a Sleeping Beauty case? Well, take the following discretized version. If it's heads, you get woken up 1,001,000 times and if's tails, you get woken up 2,001,000 times. There is no memory wiping, but empirical data from past experiments shows that you completely stop keeping track of wake-up counts after you've been woken up a thousand times. So now you've been woken up, and you know you've stopped counting. What should your credence be? This is clearly a version of Sleeping Beauty, except that instead of memory-wiping we have a cessation of keeping count, which plays the same role of being a non-rational process disturbing normal rational processes.

Oddly, though, in the sensory deprivation chamber case, I have the intuition that you should go for 1/2, even though in the original Sleeping Beauty case I've argued for 1/3. I don't have much intuition about my discretized version of the sensory deprivation chamber case.

P.s. I was thinking of blogging another Sleeping Beauty case, but it looks like LessWrong has beaten me to essentially it. (There may be a published version somewhere, too.)

William said...
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William said...

whoops, the comparison is between two factor including the coin flip.

was heads and woke having forgotten:
(1/2) * (1000001.0 - 1000)/1000001.0 = 0.4995

was tails and woke having forgotten:
(1/2) * (2000001.0 - 1000)/2000001.0 = 0.49975

So it is about 50% because the coin flip dominates the ratio.

Alexander R Pruss said...

Or you might think: There are about three million possible wakeups, and a million of them correspond to heads, so the posterior probability of heads is 1/3.

Dagmara Lizlovs said...

Alex:

Both you and LessWrong have left out Prince Charming. You can't have a Sleeping Beauty without her prince. So where's the prince? :(

Alexander R Pruss said...

He flips the coin?

Dagmara Lizlovs said...

Here's the prince:

"Man, I got these two chicks to choose from. What's a dude like me gonna do? That's it, I'll flip this here coin. Heads, I kiss Sleeping Beauty and wake up the whole palace. Tails, I go up the road with this glass slipper to Cinderella's." :-)

IanS said...

The prince’s job is the waking-up. In the second version, this is done by the experimenter, either 10001000 times or 20001000 times. This leads to the usual thirder’s argument (outlined above). In the first version, the prince seems to be missing. There is no sleeping or waking-up (at least if I understand correctly.) You know either that you have been in the chamber for less than 1 minute or for more than 1 minute. Both states will occur whether the coin lands heads or tails. So you have no reason to change your credences from ½, ½.

Do your credences even mean anything when you are in the chamber? You can’t be asked to place a bet or do anything else that depends on them.