TL;DR: SIA works only in a finite universe, as an argument for a larger universe. In the infinite universe, we can compare concentrations of observers in different regions, but it is SSA.
SIA: All other things equal, an observer should reason as if they are randomly selected from the set of all possible observers. Note that "randomly selected" is weighted by the probability of the observers existing: under SIA you are still unlikely to be an unlikely observer, unless there are a lot of them.
SSA: All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class.
Main thesis: SIA becomes SSA in the infinite universe, and the reference class in that case is the class of my exact copies.
Proof: In the infinite multiverse, there are no “possible observers”, as all possible observers actually exist somewhere, but maybe in small concentrations. (UPDATE: I discuss how infinite size of the universe generates all possible observers in comments.) Therefore, the idea of selecting from possible observers is meaningless. The selection is happening from actually existing observers. Thus, SIA turns into SSA in the multiverse.
The main difference between SIA and SSA is what they are looking at: SIA looks only at the fact that I exist at all, and SSA looks at some observables like a colour of the room in which I am located. In the infinite universe, the fact of my existence becomes uninformative: everybody exists.
Thought experiments like “God toss a coin and created one observer” are meaningless, as there is no God and there are all possible outcomes of tossing a coin. In the infinite universe, there are many of my exact copies, so we can discuss how they are distributed.
According to SIA, if there are two alternatives: (1) there are trillion observers in the universe, and (2) there are trillions of trillions of observers in the universe, – then the second hypothesis is overwhelmingly more probable. This is known as a Presumptuous Philosopher (PP) thought experiment and it is often used as a counterargument to SIA, as such predictive capability seems counterintuitive. Why does it look counterintuitive? Because it looks like a free lunch into the nature of the universe. (It is trickier with Sleeping Beauty, but let’s assume that she lives in MWI, and there are two branches, for head and for tails. In that case, probability calculations becomes straightforward.)
But the real problem of the Presumptuous Philosopher experiment is that it assumes that the universe is finite, even though that there are serious arguments that it is either infinite or a very-very large (eternal inflation). The main feature of very-very large universe is that it includes all possible observers at least once. Tegmark showed that my copy is at the distance like 10^(10^29)m.
In the very-very large universe the fact of my existence becomes non-informative, as we already know that all possible observers exist in it. Because of this, there is an alternative formulation of SIA: I am more likely to be in the part of the multiverse where most of my copies are located. There are many examples of the attempts to implement it: I am more likely to be in a simulation, in the world with MWI, in the real world vs being a Boltzmann Brain, in the world with panspermia etc. But it is not the real SIA: it is just SSA, applied to the whole multiverse where the reference class is my copies.
This infinite-world SIA gives us almost the same prediction power as the finite-world-SIA, but there are a few exceptions. Finite-world-SIA favours the hypothesis of easy abiogenesis, so life should be everywhere. Infinite-world-SIA claims that I am the universe with the easiest abiogenesis from all other universes (with some caveats) but abiogenesis still could be an extremely small probability event like 1 for 10100 planets.
Finite-SIA favours crazy hypotheses, but infinite-SIA just put me into the place where most observers-like-me live. In other words, Finite-SIA is Bayesian as it gives priors for hypothesis, and infinite-SIA is frequentist, as it is used to calculate probabilities based on the known world model.
Reference class problem.
But what about the “reference class” which is mentioned in the SSA definition, but not in the SIA definition?
Actually, there are even two reference classes in the SIA definition. The first one is the class of “all possible observers”. The second one is used to weight probabilities (as said in the definition: “Note that "randomly selected" is weighted by the probability of the observers existing”) and consists of my exact copies located in different situations. If it is zero, I do not exist. If there are many, I am located where most of my copies are located.
This probably explains why it is sometimes called “SIA+SSA”: SIA tells why I am in the tails branch in Sleeping beauty, and SSA tells where exactly I am in that branch.
Note also that “observers” in SIA still have to be “qualified observers”, who, in my opinion, are only those who already thinking about anthropics.
In other words, SIA is a SSA with a very specific choice of the reference class.
Attempt to save SIA.
The fact of being born is described in SIA as if some “empty observer” falls into the position of a particular observer, which could be in two states: existing and non-existing. Only an existing observer could think. Therefore, as I think, I am real.
SIA tries to use the opposite logic: if there is a concrete possible observer with the random name ZAQWSX, he has more chances to exist if there were more attempts to make him exist. Therefore, from the fact he exists, he could conclude that there were many attempts.
The problem with this reasoning is that it doesn’t use his name ZAQWSX and other his unique information, so any other observer will reason the same way.
Even if there was just one random observer in the universe, he can’t extract any information from his random name. Or is he?
Imagine that there are 99 red rooms and 1 green room. I am a mind which was randomly put into rooms, I am interested not in the probability of being in some of the rooms, but in the question, of how many minds were put in such rooms at all. If I am red room, I got no new information. However, if I am in the green room, I can suggest that there were many other attempts to put minds in all rooms, so at least one got into the green room.
Therefore, SIA could be saved only if I find myself in an un-typical location. Most observers can’t use this type of logic as they are typical. The presumptuous philosopher therefore should look not at the fact of his existence, but on some random coincidences in the observed physical laws.
But here there is one case where PP works: from the fact that evolution succeeds on Earth despite many risks and improbable evolutionary transitions, follows that there were billions and billions of other planets where evolution fails.
So even if we could not see stars, we will be able to guess their existence by this modified SIA.
Unexpected conclusion or a second attempt to save SIA.
We could try to save SIA by saying that the fact that “I exist at all” is an argument for an infinite universe.
(I assume that there is a way to calculate observers’ measure even in an infinite universe and an interesting idea was suggested in the article “Watchers of the Multiverse” , in which observers’ measure is calculated via eternal timelines; another idea is UDASSA.)
So, SIA proves that universe is infinite and stops here. For everything else, we use SSA with reference class of exact copies.