18 September 2005

Beautiful Sleep

Antimeta recently mentioned two famous paradoxes, those of Sleeping Beauty and the Unexpected Examination. I'd like to briefly record some thoughts on the former of these — nothing, I'm sure, that hasn't been said already, but more for posterity's sake — a longer post on the latter problem will hopefully be coming soon.

In the classic Sleeping Beauty problem, Beauty, a brilliant mathematician, signs up for an experiment to begin on Sunday at the Stanford Sleep Lab. As part of the informed consent procedures, the experimenter explains the entire proceedings:
Beauty will be put into a drug-induced sleep Sunday night. A fair coin will then be flipped. If it comes down Heads, Beauty will be awoken on Monday, asked a battery of questions, and sent on her merry way. If instead the coin lands Tails, Beauty will be awoken Monday, asked the same batter of questions, and then more drugs will wipe 24 hours from her memory and return her to sleep. She will then be awoken Tuesday, asked the questions, and sent on her merry way. The Sleep Lab room is, of course, blank, with neither calendar nor clock, and the questions are, presumably, administered by computer, so she has know immediate knowledge, upon waking, of the day, etc. The questions: "What is your credence, stated as a number between 0 and 1, that today is Monday? What is your credence that the coin landed Heads?"

I ask about credence because I don't really believe in probabilities. Or rather, the word "probability" has numerous meanings, and is used in ways to suggest a universality that can obscure its dependence on personal knowledge. (I do, incidentally, believe in Quantum Mechanics, but I interpret it a priori as a deterministic regime. Which is to say that God as an omnipotent observer sees, if you will, all the many worlds in the multiverse, and sees the universe evolving deterministically, whereas denizens of slices of that multiverse may "collapse wave functions" by "measuring things", and can experience "uncertainty", "free will", etc. I don't entirely understand how universal such experiences are.)

What should be my test for credence? I propose the following definition: If I believe a credence p for an event A, then I should be willing to wager a dollar that A is true provided that I make $1/p if I win. (I give up the dollar to play. If A is not true, then I net a dollar loss. If A is true, I net $(-1+1/p).)

Why is this rational behavior? Say that many times in my life I will be faced with a situation exactly like the one I'm in right now, and in each case I'm offered this wager. If my credence in A truly is p, then I mean by this that p of the cases when faced with this issue I'll gain $(-1+1/p) and (1-p) of the time I'll lose $1, so I'll net $p(-1+1/p) + (1-p)(-1) = $0.

So, let's amend our previous situation. Instead of asking Sleeping Beauty those questions, we'll give her a dollar wager: she wins if it's Heads and loses if it's Tails. What odds should she accept? She gets to think about this before going into the experiment (and, presumably, we'll run the experiment every week for many years, so (a) she really does want to win in the long run, and (b) all that matters is the long run, and the situation satisfies the set-up in the previous paragraph).

Well, as the experimenters, we know that half the time we'll flip heads, and half the time we'll flip tails. So each week we expect to receive $1.50 from Beauty, because she pays us a dollar each time we wake her. How much do we expect to give back? Say she accepts a payback amount of $P = 1/p (after she's paid the dollar). We'll give it to her half the time, so we'd better offer her $3 if it was in fact Heads, so that we, and hence she, break even. Similarly, every single week she will be right that it's Monday once, and she might be wrong another time. So we should give her $1.50 each time it's Monday.

Overall, with this set-up and these definitions, Beauty's credences should be 1/3 that it came up Heads, and 2/3 that it's Monday.

That in spite of this argument a case can be made for a credence of 1/2 shows, I say, that the concepts of "credence" and "probability" are nuanced. I've tried to explain, via this example, what I think "credence" should mean: given your sum knowledge, what odds should you accept on a dollar wager (pretending that exactly this situation will arise repeatedly, without dependence on how you act now). I'd like to hear other cogent understandings of the Sleeping Beauty problem, and if there is a reading for terms like "credence" and "probability" that yields the attraction 1/2 answer, please share. I expect that such a reading will weigh more on the "probability" side (our fair coin, for instance, is probably an electron which is measured to have spin in the positive x direction and then immediately thereafter its spin is measured in the y direction, and Beauty understands the question of "was it heads or tails" to be a pure QM problem), but I'm not entirely sure how to make such an interpretation viable.

1 comment:

Jack Naka said...
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