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Tuesday, March 13, 2012

Sleeping Beauty...NOT Your Childhood Fairy Tale

I've been reading about the "Sleeping Beauty Problem (or Paradox)" lately. It's actually a decade-plus-old quandary that I was aware of, but had never paid much attention to until it popped up somewhere on one of my Twitter feeds last week. Loving a good paradox, it's been rattling in my brain since.
Some folks say it reminds them of "The Monty Hall Problem" in so much as people argue vigorously for different solutions. But the Monty Hall Problem has an actual correct answer, whereas (so far as I can tell) the SBP really can be argued in two different, divergent approaches (designated as "halfers" and "thirders"). The paradox is sometimes stated in slightly variable ways, which is part of the problem, but even a fairly standard statement of it can include slight semantic pitfalls, leading to some of the disagreement. Still, more than 'Monty Hall,' the SB problem reminds me of the famous "Newcomb's Paradox" where people also tend to split two ways, and there simply is no established "right" answer.

Wikipedia states the SB problem as follows:
"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 and Tuesday. But when she is put to sleep again on Monday, she is given a dose of an amnesia-inducing drug that ensures she cannot remember her previous awakening. In this case, the experiment ends after she is interviewed on Tuesday.
Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"

Many re-state the problem to ask Sleeping Beauty, "What is the probability now for the proposition that the coin landed heads?," and some argue this word change is significant and alters the discussion. I think most people however, indeed understand the problem in terms of probabilities (the probability of heads being either 1/3 or 1/2), and so I certainly prefer thinking about it that way.

The Wikipedia entry for the problem is here:

http://en.wikipedia.org/wiki/Sleeping_Beauty_problem

And here are some more links discussing it (warning though, they may cause your head to pound ;-)):

http://blog.tanyakhovanova.com/?p=356 (arguing for the 1/3 answer)

http://barryispuzzled.com/zbeauty.htm (arguing for the 1/2 answer)

http://meteuphoric.wordpress.com/2011/01/09/against-the-hybrid/

…and discussion from a "freethought forum" here:

http://www.freethought-forum.com/forum/showthread.php?t=3120

Finally, if that's not enough for you, you glutton-for-punishment, more links here:

http://www.anthropic-principle.com/?q=resources/preprints#pre5

3 comments:

Sol Lederman said...

Shecky, I love this! I'd never heard of the SBP. Will have to read up on it.

"Shecky Riemann" said...

Hi Sol, thanks... yeah, I love that I can read one set of arguments and find them convincing, only to read an opposing set of arguments and be swayed back to a different view... over & over again! (I lean toward 1/2 as the strictly mathematical correct answer, but more subtly in terms of "credence" or belief 1/3 makes a lot of sense to me... or that's what I think at this particular moment!)

JeffJo said...

Just like in the Monty Hall Problem, you must learn to differentiate between the probability of something occurring, or having occurred with no information about what did; and the probability it did occur based on some incomplete information about what did. The two are not the same. And the controversy over credence vs. probability is just an attempt to rationalize calling them the same thing.

In the Monty Hall Problem, the chances that the door you would switch to has a goat started out at 1/2. But because Monty Hall could have opened it in that case, but not if it had the car, the chances become smaller. Quite literally, it is the possibility of something that could have happened, but you know didn’t, that changes the answer.

In the Sleeping Beauty Problem there are four (not three) combinations of a day and coin result that can occur. But only three can result in Sleeping Beauty observing them. And it is the possibility of what you know didn’t happen that changes the answer from the 1/2 it was before the experiment, to the 1/3 that it is when SB is awakened.

And if you doubt it, make a simple change: flip a second coin. If it lands on heads, the experiment is the same. If it lands on tails, awaken SB the same number of times as you would have otherwise, but make Monday be the day she might not be awakened. Several facts become clearer now: (1) My four combinations are all possible, by whatever definition you want to use. (2) Regardless of what day it is, the probability the first coin landed on heads is half the probability it landed tails. SB will always be awakened on each day if it was heads, but only half of the time on each day if it landed tails. SB's credence fro heads must be 1/3. (3) Since the two variations that are based on the second coin are functionally equivalent, and one is identical to the original experiment, the answer cannot be different.