So what if we are uncertain about the size of the universe (so that its size depends on which possible world we are in)? Then we are faced with the same question as before: Should we treat finding ourselves in bigger universes as more probable a priori, or not?
Formally, the question we face is, if we have a prior P0 over possible worlds, what should our prior over (possible world, space-time-Everett location) pairs be?
Physical self-sampling without self-indication. P((w,x)) = P0(w) / number of possible locations in world w
Physical self-sampling with physical self-indication. P((w,x)) = alpha P0(w), where alpha is a normalization constant (alpha = 1 / Sum_w’. P0(w’) number of possible locations in world w’)
(As before, we may want to weigh Everett branches in the obvious way.) Both of these definitions give us the weak principle of self-indication (defined in the previous comment), since they agree with the previous comment’s definition when all possible worlds contain the same number of locations. So they both support thirding in Sleeping Beauty.
But which of the definitions should we adopt? Note that sampling without self-indication has the property that P(w) = P0(w), i.e., before we condition on any evidence (including the fact that we are conscious observers), the probability of finding ourselves in world w is exactly the probability of that world, according to P0. On the face of it, this sounds exactly like what we mean by having a prior P0 over the possible worlds.
I think we may mean different things with P0 depending on how we arrive at P0, though. But for the moment, let me note that while the principle of weak self-indication forces me to accept the presumptuous philosopher’s position in both the Case of the Twin Stars and the Case of the Death Rays, I may still have a good reason to reject the conclusion that the cosmos is infinite with probability one.
So what if we are uncertain about the size of the universe (so that its size depends on which possible world we are in)? Then we are faced with the same question as before: Should we treat finding ourselves in bigger universes as more probable a priori, or not?
Formally, the question we face is, if we have a prior P0 over possible worlds, what should our prior over (possible world, space-time-Everett location) pairs be?
(As before, we may want to weigh Everett branches in the obvious way.) Both of these definitions give us the weak principle of self-indication (defined in the previous comment), since they agree with the previous comment’s definition when all possible worlds contain the same number of locations. So they both support thirding in Sleeping Beauty.
But which of the definitions should we adopt? Note that sampling without self-indication has the property that P(w) = P0(w), i.e., before we condition on any evidence (including the fact that we are conscious observers), the probability of finding ourselves in world w is exactly the probability of that world, according to P0. On the face of it, this sounds exactly like what we mean by having a prior P0 over the possible worlds.
I think we may mean different things with P0 depending on how we arrive at P0, though. But for the moment, let me note that while the principle of weak self-indication forces me to accept the presumptuous philosopher’s position in both the Case of the Twin Stars and the Case of the Death Rays, I may still have a good reason to reject the conclusion that the cosmos is infinite with probability one.