Both you and the other reply refer to counting universes/branches, which is obviously a term of art that I’m not understanding. Where I talk about sampling successor universes, are you just emphasizing that we’re drawing from a continuous probability distribution instead of a discrete one, or is there something else?
Actually, it's a hand-wavy term that is not well-defined, because "universe" and "branch" and "world" are not well-defined.
> sampling successor universes
The problem with this is that in order to sample something you have to specify the procedure by which you are going to do the sampling because your results are going to depend on that procedure. So even leaving aside the fact that the thing that you're trying to sample is not even well-defined, there are other issues.
Sampling finite sets is straightforward, but for infinite sets it gets tricky [1], and for infinite sets without a total ordering it gets very tricky. Consider, for example, generating a random complex number. There are at least two plausible ways to do this:
1. Generate two random reals x and y, and combine them to produce the random complex number x + iy.
2. Generate two random reals, r and theta and combine them to produce the random complex number r * (sin(theta) + i cos(theta)).
Depending on which of these methods you choose, you'll get different-looking distributions. Neither one is "correct".
If you're going to talk about sampling worlds you have all these difficulties in spades because worlds are rays in Hilbert spaces, i.e. vector spaces with an infinite number of dimensions. There are an infinite number of ways to generate random distributions over a Hilbert space. The trick is to make an argument for why one particular way is better than all the others.
The obvious argument is to present a way of sampling that reproduces the Born rule and argue that it's better because it reproduces the experimental results. But in order to not be begging the question, this sampling procedure can't have the Born rule hidden within it. That is what Wallace and Deutsch claim to have done, but which I (and many others) dispute.
Thanks; I had assumed that drawing a sample was a fundamental operation on a well-defined probability distribution, and that existing QM tools were sufficient to define that distribution, if not distinguish between the various interpretations.
