Okay, after reading several of Nick Bostrom’s papers and mulling about the problem for a while, I think I may have sorted out my position enough to say something interesting about it. But now I’m finding myself suffering from a case of writer’s block in explaining it, so I’ll try to pull a small-scale Eliezer and say it in a couple of hiccups, rather than one fell swoop :-)
I have been significantly wrong at least twice in this thread, the first time when I thought everybody was reasoning from the same definitions as me, but getting their math wrong, and the second time when I said I held my view because I was “pretty sure I [was] applying my probability theory right”. I had an intuition and a formal argument, but then I found that the two disagree in some edge cases, and I decided to retain the intuition, so my formal argument was not the solid rock I thought it was. All of which is a long-winded way of saying, it’s about time that I concede that I may still be wrong about this, and if so, please do help me figure it out...
We all seem to agree that the issue depends on whether we accept self-indication, and that self-indication is equivalent to being a thirder in the Sleeping Beauty problem. When I first learned about this problem from Robin’s post, I was very convinced that the halfer view was right—to the tune of having been willing to bet money on it—for about fifteen minutes. Then I thought about something like the following variation of it:
Beauty is put to sleep on Sunday, and a fair coin is tossed. Beauty is awakened twice, once on Monday and once on Tuesday; in between, she is given an amnesia-inducing drug, so that when she wakes up, she cannot tell whether she has been woken up before. One minute after Beauty wakes up, a light flashes. If it is Tuesday, and the coin came up heads, the light is red; otherwise, it is blue.
When Beauty wakes up, before the light flashes, what is her subjective probability that (h1) the coin came up heads, and it’s Monday; (h2) heads, Tuesday; (t1) tails, Monday; (t2) tails, Tuesday?
I cannot conceive of a reason not to assign the probability 1⁄4 to each of these propositions, and in my opinion, when Beauty sees the light flash red, she must update her subjective probability in the obvious way (or the notion of subjective probability no longer makes much sense to me). Then, of course, after seeing the light flash blue, Beauty’s probability that the coin fell heads is 1⁄3.
Short of assigning special ontological status to being consciously awake, I don’t see a way to distinguish between the original Sleeping Beauty and my variation after the light flashes blue, so I’m a thirder now. My new view is that observing the random variable (color=blue) can change my probability in non-mysterious ways, so observing the random variable (awake=yes) can, too.
In his paper on the problem, Nick argues for a “solution” that would apply to my version, too. He would reject my view of how Beauty must update her probabilities if she sees a blue light. His argument goes something like this:
What I really need to consider is all of Beauty’s observer-moments in all possible worlds; Beauty has a prior over these moments, considers the evidence she has for which moment she is in, and does a Bayesian update. The moment when Beauty wakes up is different from the moment when the light flashes, so she needs to consider at least eight possible moments: (h1-) heads, Monday, she wakes up; (h1+) heads, Monday, the light flashes; and so on. Nothing in the axioms of probability theory requires the probability of (h1+) to be related in any way to the probability of (h1-)! In fact, Nick would argue, we should simply assign probabilities like this:
p(xx- | h1- \/ h2- \/ t1- \/ t2-) = 1⁄4 (for xx in {h1,h2,t1,t2})
p(h1+ | h1+ \/ t1+ \/ t2+) = 1⁄2
p(xx+ | h1+ \/ t1+ \/ t2+) = 1⁄4 (for xx in {t1,t2})
I agree that this is formally consistent with the axioms of probability, but in order for Beauty to be rational, in my opinion she must still update her probability estimate in the “normal” way when the light flashes blue. Nick’s approach strikes me as saying, “I’m a completely new observer-moment now, why should I care about my probability estimates a minute ago?” If our formalism allows us to do that, I think our formalism isn’t strong enough. In this case, I’d require that
--i.e., before conditioning on the actual colors she sees, Beauty’s probability estimates when the light flashes must be the same as when she wakes up. I don’t know how well this generalizes, but if we accept it in this case, it blocks Nick’s proposal.
Anybody here who finds Nick’s solution intuitively right?
Okay, after reading several of Nick Bostrom’s papers and mulling about the problem for a while, I think I may have sorted out my position enough to say something interesting about it. But now I’m finding myself suffering from a case of writer’s block in explaining it, so I’ll try to pull a small-scale Eliezer and say it in a couple of hiccups, rather than one fell swoop :-)
I have been significantly wrong at least twice in this thread, the first time when I thought everybody was reasoning from the same definitions as me, but getting their math wrong, and the second time when I said I held my view because I was “pretty sure I [was] applying my probability theory right”. I had an intuition and a formal argument, but then I found that the two disagree in some edge cases, and I decided to retain the intuition, so my formal argument was not the solid rock I thought it was. All of which is a long-winded way of saying, it’s about time that I concede that I may still be wrong about this, and if so, please do help me figure it out...
We all seem to agree that the issue depends on whether we accept self-indication, and that self-indication is equivalent to being a thirder in the Sleeping Beauty problem. When I first learned about this problem from Robin’s post, I was very convinced that the halfer view was right—to the tune of having been willing to bet money on it—for about fifteen minutes. Then I thought about something like the following variation of it:
I cannot conceive of a reason not to assign the probability 1⁄4 to each of these propositions, and in my opinion, when Beauty sees the light flash red, she must update her subjective probability in the obvious way (or the notion of subjective probability no longer makes much sense to me). Then, of course, after seeing the light flash blue, Beauty’s probability that the coin fell heads is 1⁄3.
Short of assigning special ontological status to being consciously awake, I don’t see a way to distinguish between the original Sleeping Beauty and my variation after the light flashes blue, so I’m a thirder now. My new view is that observing the random variable (color=blue) can change my probability in non-mysterious ways, so observing the random variable (awake=yes) can, too.
In his paper on the problem, Nick argues for a “solution” that would apply to my version, too. He would reject my view of how Beauty must update her probabilities if she sees a blue light. His argument goes something like this:
What I really need to consider is all of Beauty’s observer-moments in all possible worlds; Beauty has a prior over these moments, considers the evidence she has for which moment she is in, and does a Bayesian update. The moment when Beauty wakes up is different from the moment when the light flashes, so she needs to consider at least eight possible moments: (h1-) heads, Monday, she wakes up; (h1+) heads, Monday, the light flashes; and so on. Nothing in the axioms of probability theory requires the probability of (h1+) to be related in any way to the probability of (h1-)! In fact, Nick would argue, we should simply assign probabilities like this:
p(xx- | h1- \/ h2- \/ t1- \/ t2-) = 1⁄4 (for xx in {h1,h2,t1,t2})
p(h1+ | h1+ \/ t1+ \/ t2+) = 1⁄2
p(xx+ | h1+ \/ t1+ \/ t2+) = 1⁄4 (for xx in {t1,t2})
I agree that this is formally consistent with the axioms of probability, but in order for Beauty to be rational, in my opinion she must still update her probability estimate in the “normal” way when the light flashes blue. Nick’s approach strikes me as saying, “I’m a completely new observer-moment now, why should I care about my probability estimates a minute ago?” If our formalism allows us to do that, I think our formalism isn’t strong enough. In this case, I’d require that
p(xx- | h1- \/ h2- \/ t1- \/ t2-)
= p(xx+ | h1+ \/ h2+ \/ t1+ \/ t2+)
--i.e., before conditioning on the actual colors she sees, Beauty’s probability estimates when the light flashes must be the same as when she wakes up. I don’t know how well this generalizes, but if we accept it in this case, it blocks Nick’s proposal.
Anybody here who finds Nick’s solution intuitively right?