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.

Grabby and visible aliens

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^[1]. 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.

Rare Earth hypotheses

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 $T_0$ say that life can exist on any planet with rate $\rho$, while the rare Earth hypothesis, $T_1$, 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 $T_1$ as $T_0$ with a different prior on $\rho =pr$ (induced from the priors on $p$ and $r$), but both these updates affect $\rho$.

Now, $T_0$ and $T_1$ 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.

Independent aliens

Suppose that theory $T_2$ posits that there are aliens in, say, gas giants, whose existence is independent from ours. Visible gas giant alien civilizations exist at a rate ${\rho }_g$, while visible life on rocky planets exist at a rate $\rho$.

Then the anthropic update boosts $\rho$ only, while the Fermi observation penalises $\rho$ and ${\rho }_g$ equally (if we make the simplifying assumption that gas giants are as common as terrestrial planets).

This gives a differential boost to $\rho$ over ${\rho }_g$, 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 $\rho$ and ${\rho }_g$, then after updating, we get:

$$\rho =\frac{2}{N+2},{\rho }_g=\frac{1}{N+1}.$$

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 $\rho$ and ${\rho }_g$.

So, some differential effect due to anthropics, but not a strong one, at least for uniform priors, and not one that grows.

In the cosmic zoo

Let $T_3$ 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 $T_3$ 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:

1. In the zoo hypothesis, aliens are hiding themselves from us. This is close to a "Descartes's demon" hypothesis, in that powerful entities are acting to feed us erroneous observations. Pure Descartes's demon hypotheses are not differentially boosted by anything, since they explain nothing (once you've posited a demon, you also have to explain why we see what we think we see). The zoo hypothesis is not quite as bad - "keep everything hidden" is more likely than other ways aliens could be messing with our observations. Still, it should be a low prior.
2. Though the Fermi observation doesn't downgrade the zoo hypothesis directly, the more carefully we observe the universe, the more unlikely it becomes, since the aliens would have to work harder to conceal any evidence.
3. Conversely, the more visible we become, the less likely the zoo hypothesis becomes, because we have to explain why the zookeepers haven't intervened to keep us concealed (if we suppose that these aliens are powerful enough to intercept light and other signals between the stars, then we're very close to the Descartes demon territory). Once we successfully launched replicating AIs to the stars, then we'd be pretty sure the zoo hypothesis was wrong.

Time enough for aliens

So far, we've neglected time in the equation, talking about a rate $\rho$ that was per planet, but not stretched over time. But consider theory $T_4$: 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, $T_4$ can get boosted relative to them - and the more dead planets we observe or infer, the stronger the relative boost is.

Now, $T_4$ 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 $T_4$, 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 $T_5$ that posit that advanced life started much earlier than the present day, pay a much higher price via the Fermi observation.