Imagine a countably infinite multiverse where there are no spatial relations between the different universes, but it makes sense to talk of a common time sequence measured by the duration of time elapsed from the beginning of a universe. Suppose all the universes start out the same way: they each consist of a closed box containing an unstable particle with a one-year half-life that decays into a stable particle. The only change there is is the decay of the particle.
Every year "half" of these particles decay. But now notice an oddity: from the second year on, there is a sense in which there are no global change at all. No matter how many particles have decayed, the global picture will forever be this: a multiverse consisting of universes infinitely many of which consist of a box with an unstable particle and infinitely many consist of a box with a stable particle. In other words, there is no qualitative change in the multiverse. Interesting and somewhat paradoxical: it looks like reality is unchanging and yet there is change. We capture the change when we keep track of non-qualitative features: this box had an unstable particle, and now this same box has a stable particle.
Now let's add one person to each universe. The person doesn't get to look inside the box, but knows the general setup. Each year, the person's probability that her box contains an unstable particle goes down by a half.
Lesson 1: self-locating probabilities cannot be read off of the qualitative features of the present state of the universe--either history or numerical identity matters. (This is the same lesson that Ian has been pushing me to learn from an earlier post.)
Now, in the above story, the universe started out with all the particles in the unstable state. But what if instead the universe started out with infinitely many of the particles in the stable state and infinitely many in the unstable state? What can we say about each person's probability that her box contains an unstable particle? Each year after the first, P(unstable now | unstable a year ago) = 1/2. But what about the unconditional probability? How likely is it after, say, one year that the particle in one's box is unstable? Well, it's half of the probability that it was unstable to begin with.
But what is the probability that it was unstable to begin with? Well, if we had a bit of a backstory, we could answer that. Let's say that God creates each multiverse, and he flips a fair coin as he gets to each, and puts in a stable particle on heads and an unstable one on tails. In that case, the probability that the particle in the box was unstable to begin with was 1/2. But if God instead rolled a die, and put in a stable particle on six and an unstable one otherwise, the probability of initial instability is 5/6. In other words, the probability of initial instability depends on backstory. But what if there is no backstory at all? What if this multiverse came into existence ex nihilo for no cause at all? It is tempting to assign probability 1/2 to instability. But why that? Why not, instead, 1/6 or 5/6 or something else? The choice seems arbitrary and the honest thing to say, I think, is just that there is no meaningful probability.
Lesson 2: Probability of initial conditions requires some sort of structure. A chancy causal structure will do. Perhaps a spatial structure could do--if the boxes were in a single spaced, arranged on a line, alternating between stable and unstable, it might be reasonable to say that the probability that one has instability is 1/2. Might be, but I'm not sure.
But of course probabilities in subsequent years will depend on initial probabilities. If we cannot say anything about initial probabilities without more structure, we can't say anything about probabilities in subsequent years, except conditionally.
Does anything in Lesson 2 hang on us working in a multiverse? I doubt it.