We updated on the fact that we exist. SSA does this a little too: specifically, the fact that you exist means that there is at least one observer. One way to look at it is that there is initially a constant number of souls that get used to fill in the observers of a universe. In this formulation, SIA is the result of the normal Bayesian update on the fact that soul-you woke up in a body.
Suppose there is a 2^(-n) chance of universe U_n with n people for n > 0. Initially, there’s nothing paradoxical about this. SIA converges. But look at the evidence you get from existing. Call that E.
P(U_n|E) = knP(U_n) for some k
P(U_n|E) = P(U_n&E)/P(E)
P(U_n|E) < P(U_n)/P(E)
P(E) < P(U_n)/P(U_n|E)
P(E) < P(U_n)/(knP(U_n))
P(E) < 1/kn
Since k is constant and this is true for all n, P(E) = 0
So, existence is infinitely unlikely? Or we must assume a priori that the universe definitely doesn’t have more than n people for some n? Or P(U_n&E) is somehow higher than P(U_n)?
Right. In a sense, P(E) is one over the number of possible people in the universe (scaled by how much of configuration space ‘you’ are).
Observing your existence only changes your probabilities if your nonexistence was also, causally, an option. In order for there to be an infinite number of possible people in the universe, but only this exponential prior distribution, the probability that any given chunk of stuff is ‘you’ has to go to zero.
This kinda confused me, I think because P(E) does not represent what I’d colloquially expect—I’m pretty sure now that it’s a sampling probability, not a global probability.
the probability that any given chunk of stuff is ‘you’ has to go to zero.
I don’t see why. If there’s an x% chance of a given chunk of stuff in universe U_1 being you, and there’s a 50% chance of universe U_1, then there should at least be an x/2% chance of a given chunk of stuff being you, right? And those calculations don’t actually use that it’s an exponential decrease. It could have been P(U_n) = 1/Σ(n) and it would still apply.
I’m not sure how you’re thinking of this, but think about what’s going on in jesscat’s “souls” toy picture. In order for U_n to be possible, there must be at least n souls. Since U_n exists for every n, there must be an infinite number of souls—and therefore zero chance that when we pick out exactly one soul for U_1, yours is the soul that gets chosen. Therefore, ‘your’ probability of existing in U_1 is 0.
This almost works like if each ‘soul’ was a configuration of a person, and you were one particular configuration—except U_n (and SIA) don’t specify that each person has to be unique. Instead, it’s more analogous to a particular configuration of a particular chunk of matter—that’s one way to put in uniqueness.
The vast majority of possible souls live in chaotic universes. Under this theory, rather than just having a random experience like a Boltzmann brain, you almost certainly have no experience. But having a sensible experience is still astronomically low.
We updated on the fact that we exist. SSA does this a little too: specifically, the fact that you exist means that there is at least one observer. One way to look at it is that there is initially a constant number of souls that get used to fill in the observers of a universe. In this formulation, SIA is the result of the normal Bayesian update on the fact that soul-you woke up in a body.
Suppose there is a 2^(-n) chance of universe U_n with n people for n > 0. Initially, there’s nothing paradoxical about this. SIA converges. But look at the evidence you get from existing. Call that E.
P(U_n|E) = knP(U_n) for some k
P(U_n|E) = P(U_n&E)/P(E)
P(U_n|E) < P(U_n)/P(E)
P(E) < P(U_n)/P(U_n|E)
P(E) < P(U_n)/(knP(U_n))
P(E) < 1/kn
Since k is constant and this is true for all n, P(E) = 0
So, existence is infinitely unlikely? Or we must assume a priori that the universe definitely doesn’t have more than n people for some n? Or P(U_n&E) is somehow higher than P(U_n)?
Right. In a sense, P(E) is one over the number of possible people in the universe (scaled by how much of configuration space ‘you’ are).
Observing your existence only changes your probabilities if your nonexistence was also, causally, an option. In order for there to be an infinite number of possible people in the universe, but only this exponential prior distribution, the probability that any given chunk of stuff is ‘you’ has to go to zero.
This kinda confused me, I think because P(E) does not represent what I’d colloquially expect—I’m pretty sure now that it’s a sampling probability, not a global probability.
I don’t see why. If there’s an x% chance of a given chunk of stuff in universe U_1 being you, and there’s a 50% chance of universe U_1, then there should at least be an x/2% chance of a given chunk of stuff being you, right? And those calculations don’t actually use that it’s an exponential decrease. It could have been P(U_n) = 1/Σ(n) and it would still apply.
I’m not sure how you’re thinking of this, but think about what’s going on in jesscat’s “souls” toy picture. In order for U_n to be possible, there must be at least n souls. Since U_n exists for every n, there must be an infinite number of souls—and therefore zero chance that when we pick out exactly one soul for U_1, yours is the soul that gets chosen. Therefore, ‘your’ probability of existing in U_1 is 0.
This almost works like if each ‘soul’ was a configuration of a person, and you were one particular configuration—except U_n (and SIA) don’t specify that each person has to be unique. Instead, it’s more analogous to a particular configuration of a particular chunk of matter—that’s one way to put in uniqueness.
The vast majority of possible souls live in chaotic universes. Under this theory, rather than just having a random experience like a Boltzmann brain, you almost certainly have no experience. But having a sensible experience is still astronomically low.