Researcher at the Center on Long-Term Risk. Located in London.
SIA implies a different conclusion. To predict your observations under SIA, you should first sample a random universe proportional to its population, then sample a random observer in that universe. The probabilities of observing each index are the same conditional on the universe, but the prior probabilities of being in a given universe have changed.
We start with 1000:1 odds in favor of the 1-trillion universe, due to its higher population. Upon observing our sub-1-billion index, we get a 1000:1 update in favor of a 1-billion universe, as with SIA. These exactly cancel out, leaving the probability of each universe at 50%.
I found this is a somewhat confusing/counterintuitive way to explain SIA. (Because why start out with 1000:1 odds when it is stated that the two universes are equally likely a priori?)
How I'd explain it: Suppose I want to put a credence on the hypothesis T Trillion observer universe, starting from a 50% prior on each of the the two hypotheses. I now observe my index being, say, 100. According to my prior, what’s the expected fraction of observers that observe having index 100 who would be correct to believe T? It’s 50%.
That phrasing sounds right, yeah.
I wrote expected fraction in the previous comment in order to circumvent the requirement of both universes existing simultaneously. But I acknowledge that my intuition is more compelling when assuming that they all exist, or (in Sleeping Beauty or God's extreme coin toss) that the experiment is repeated many times. Still, it seems odd to expect an inconsistency between the "it just happened once" and the "it happens many times" cases..