When updating the probability of life across the universe, there are two main observations we have to build on:
Anthropic update: we exist on Earth.
Fermi observation: (we exist on Earth and) we don't see any aliens.
I'll analyse how these two observations affect various theories about life in the universe. In general, we'll see that the anthropic update has a pretty weak effect, while the Fermi observation has a strong effect: those theories that benefit most are those that avoid the downgrade from the Fermi, such as the Zoo hypothesis, or the "human life unusually early" hypothesis.
I've argued that an anthropic update on our own existence is actually just a simple Bayesian update; here I'll explain what that means for our updates.
This paper talks about grabby aliens, who would expand across the universe, and stop humans from evolving (if they reached Earth before now). As I've argued, "we exist" and "we have not observed X" are statements that can be treated in exactly the same way. We can combine them to say "there are no visible aliens anywhere near", without distinguishing grabby aliens (who would have stopped our existence) from visible-but-not grabby aliens (who would have changed our observations).
Thus the Fermi observation is saying there are no grabby or visible aliens nearby. Recall that it's so comparatively easy to cross between stars and galaxies, so advanced aliens would only fail to be grabby if they coordinated to not want to do so.
Some theories posit that life requires a collection of conditions that are very rarely found together.
But rare Earth theories don't differ much, upon updates, from "life is hard" hypotheses.
For example, suppose T0 say that life can exist on any planet with rate ρ, while the rare Earth hypothesis, T1, says that life can exist on Earth-like planets with rate p, while Earth-like planets themselves exist with rate r.
But neither the Fermi observation nor the anthropic update will distinguish between these theories. The Fermi observation posits that there are no visible aliens close to Earth; the anthropic updates increases the probability of life similar to us. We can see T1 as T0 with a different prior on ρ=pr (induced from the priors on p and r), but both these updates affect ρ.
Now, T0 and T1 can be distinguished by observation (seeing dead planets with Earth-like features) or theory (figuring out what is needed for life). Anything that differentially change p and r. But neither the anthropic update nor the Fermi observation do this.
Suppose that theory T2 posits that there are aliens in, say, gas giants, whose existence is independent from ours. Visible gas giant alien civilizations exist at a rate ρg, while visible life on rocky planets exist at a rate ρ.
Then the anthropic update boosts ρ only, while the Fermi observation penalises ρ and ρg equally (if we make the simplifying assumption that gas giants are as common as terrestrial planets).
This gives a differential boost to ρ over ρg, but the effect can be mild. If we assume that there are N gas giants and N terrestrial planets in the milky way, and start with a uniform prior over both ρ and ρg, then after updating, we get:
If the rate of gas giant alien civilizations is semi-dependent on our own existence - maybe we both need it to be easy for RNA to exist - then there will be less of a difference in the update for ρ and ρg.
So, some differential effect due to anthropics, but not a strong one, at least for uniform priors, and not one that grows.
Let T3 be a cosmic zoo hypothesis. It posits that there may be a lot of aliens, but they have agreed - or been coerced - into hiding themselves, so as not to contaminate human development (or some other reason).
Then T3 gets a boost from the anthropic update, and no penalty from the Fermi observation. Since most theories get a big downgrade from the Fermi observation, this can raise its probability quite a lot relative to other theories.
A few caveats, however:
So far, we've neglected time in the equation, talking about a rate ρ that was per planet, but not stretched over time. But consider theory T4: advanced life starts appearing around 13.77 billion years after the Big Bang, but not before.
This theory might be unlikely, but it gets a mild boost from anthropics (since it's compatible with our existence) and avoids the downgrade from the Fermi observation (since it says there are no visible aliens - yet).
Since that downgrade has been quite powerful for most theories, T4 can get boosted relative to them - and the more dead planets we observe or infer, the stronger the relative boost is.
Now, T4 may seem unlikely, since the Earth is a late planet among the Earth-like planets: "Thus, the average earth in the Universe is 1.8 ± 0.9 billion years older than our Earth". But there are some theories that make more plausible T4, such as some versions of panspermia. Specifically, if we imagine that life had to go through several stages, on several planets - maybe RNA/DNA was the result of billions of years of evolution on a planet much older than the Earth, and was then spread here, where it allowed another stage of evolution.
Conversely, theories T5 that posit that advanced life started much earlier than the present day, pay a much higher price via the Fermi observation.
The grabby alien paper uses "loud" to designate aliens that "expand fast, last long, and make visible changes to their volumes". Visible aliens are more general; in particular, they need not expand (though this may make them less visible). ↩︎
There could be two types of Zoos. Using human analogy, city zoo and wild-life reserve. City zoo has a few animals under very tight control, and they can live in a perfect simulation of the intact world, like fishes in tank. Wild-life reserve has more animals, but less protected from wrong observations. E.g. hunting grounds.
The second type is more probable based on anthropics, as it includes many more observers, and if we are in a zoo, it is probably of the second type. It may explain UFO observation, or, if we discard UFOs, it is an argument againt zoo.